1. Introduction
The Hawkes process is a general counting process, denoted by N, that is often used to model the occurrence of random events that display temporal clustering behaviour [Reference Hawkes19]. The complete intensity function of
$N_t$
is given by
\begin{align*}\lambda_t = \nu + \int_{-\infty}^{t-} g(t-s) \,\mathrm{d} N_s , \end{align*}
where
$\nu >0$
is known as the baseline intensity, g is known as the excitation kernel and is responsible for describing the rate of occurrence of excited events. Since being proposed, the Hawkes process and its extensions have seen a diverse range of applications in areas such as neurosciences [Reference Reynaud-Bouret, Rivoirard and Tuleau-Malot35], limit order book modelling and trade arrival times for stocks [Reference Clinet and Yoshida7, Reference Da Fonseca and Zaatour12], earthquake modelling [Reference Ogata33] and prediction [Reference Vere-Jones41], social network interaction modelling [Reference Crane and Sornette9], and infectious disease transmission modelling in epidemiology [Reference Escobar15, Reference Mohler and Short30]. The Hawkes process has also seen applications in conjunction with modern machine learning techniques [Reference Hansen, Reynaud-Bouret and Rivoirard17, Reference Mei and Eisner29, Reference Zhang, Lipani, Kirnap and Yilmaz42].
In the traditional setting, the classical Hawkes process assumes the excitation kernel takes an exponential form, that is,
$g(t) = \alpha {\mathrm{e}}^{-t\beta}$
,
$\beta >\alpha>0$
. As a consequence of the memoryless property of the exponential distribution, the complete intensity process
$\lambda_t$
is Markovian [Reference Oakes31] and enjoys certain ergodic-type properties [Reference Abergel and Jedidi1]. Initial works by Hawkes [Reference Hawkes18–Reference Hawkes20] and Hawkes and Oakes [Reference Hawkes and Oakes21] examined the probabilistic properties of the Markovian Hawkes process, whilst Ozaki [Reference Ozaki34] proposed a maximum likelihood estimation procedure for the process. Furthermore, in a more general setting, for point processes that are stationary and ergodic, Ogata [Reference Ogata32] established consistency and asymptotic normality of the maximum likelihood estimators under certain regularity conditions. Utilising techniques from ergodic theory for Markov processes, recent work by Clinet and Yoshida [Reference Clinet and Yoshida7] establishes ergodicity of the classical multivariate Hawkes process and a quasi-likelihood procedure is realised, with consistency and asymptotic normality of the quasi-MLE.
While the aforementioned contributions mostly assume a classical Hawkes process setup, the assumption of an exponential g can be restrictive and unrealistic for modelling purposes. For example, the infectiousness of COVID-19 is found to be causally driven by the incubation period [Reference Ferretti, Ledda, Wymant, Zhao, Ledda, Abeler-Dörner, Kendall, Nurtay, Cheng, Ng, Lin, Hinch, Masel, Kilpatrick and Fraser16], while another study found the median incubation period was estimated to be 5.1 days (95% CI, 4.5 to 5.8 days) [Reference Lauer, Grantz, Bi, Jones, Zheng, Meredith, Azman, Reich and Lessler28]. This suggests that, informally speaking, the peak infectiousness of someone who has contracted COVID-19 occurs at a delay after being infected, so a Gamma or Weibull density function with shape parameter
$k >1$
as a candidate for g may provide a more appropriate fit compared to an exponential one, which was used in [Reference Escobar15]. In other areas of research, the Hawkes process with a power law kernel has been used to model human activities [Reference Crane and Sornette9, Reference Sornette, Deschâtres, Gilbert and Ageon38, Reference Vázquez and Oliveira40] and earthquakes [Reference Helmstetter and Sornette22]. Despite their popularity, some of their main probabilistic properties are yet to be established. The key step in [Reference Clinet and Yoshida7] to establish consistency and asymptotic normality of the quasi-MLE relies on the ergodic property of the intensity function of the classical Hawkes process, which in turn allows the identification of, and inference to be done on, the ergodic limit of the quasi-likelihood function. However, inference of a similar nature is substantially more difficult to replicate when g is non-exponential, since the point process is non-Markovian. While a likelihood-based procedure may still be implemented to estimate the parameters of non-Markovian Hawkes processes, fundamental asymptotic results of the MLE, e.g. consistency and asymptotic normality, for such processes are still absent from the current literature.
There are existing probabilistic results on non-Markovian Hawkes processes. For example, recent work by Cui et al. [Reference Cui, Hawkes and Yi10] presented a procedure to obtain moments for a certain non-Markovian Hawkes process. Such a procedure was applied in [Reference Cui, Wu and Yin11] to retrieve the moments of the Hawkes process with a Gamma decaying kernel. Costa et al. [Reference Costa, Graham, Marsalle and Tran8] used renewal arguments to establish limit theorems for Hawkes process with excitation kernels that are signed and have bounded support. Bordenave and Torrisi [Reference Bordenave and Torrisi3] and Zhu [Reference Zhu43] studied large deviations type results for
$N_t / t$
, where
$N_t$
is a Hawkes process with a general excitation kernel, suggesting that the limit of the distribution of
$N_t/t$
, after some appropriate transformation and scaling, exists while the rate of convergence depends on the model parameters. As it happens, in both [Reference Bordenave and Torrisi3] and [Reference Zhu43] it is claimed that if the Hawkes process is ergodic, then it has finite mean intensity
$\mathbb{E}[N_1] - \mathbb{E} [N_0] =m$
and
\begin{equation*} \lim_{t \rightarrow \infty} \dfrac{N_t - N_0}{t} = m\end{equation*}
almost surely [Reference Zhu43, Lemma 1]. While surveying the literature, we observed that various representations of ergodicity were adopted depending on the context and underlying stochastic process of interest. For example, the two-part textbook by Daley and Vere-Jones [Reference Daley and Vere-Jones13, Reference Daley and Vere-Jones14] gave two definitions of ergodicity of point processes. Definition 3.5.II of [Reference Daley and Vere-Jones13] coincides with the property stated in equation (1.3) of [Reference Zhu43] and equation (4) of [Reference Bordenave and Torrisi3], where a stationary point process with finite mean density
$m = \mathbb{E} [N_1 - N_0] < \infty$
is ergodic when
$N_x/x \overset{\text{a.s}}{\rightarrow} m$
, while Definition 12.3.I of [Reference Daley and Vere-Jones14] requires a detailed prerequisite in measure theory. Ogata [Reference Ogata32] assumed the point process is ergodic; however, no formal definition of ergodicity was given. Despite the various statements of ergodicity in the literature, they all share an essential property that when a stochastic process is ergodic, certain statistical properties can be deduced from a single and sufficiently long random sample. For the purpose of inference on the likelihood function of self-exciting point processes, ergodicity of the intensity process is more relevant. For this reason, we adopt the definition of ergodicity in [Reference Clinet and Yoshida7] (see also Definition 2.1 below).
In this paper, by imposing minimal regularity conditions on the model, we establish ergodicity of the intensity process of the stationary Hawkes process with a constant baseline intensity and a non-exponential kernel. Ergodicity is achieved by first noting that there exists a stationary Poisson cluster process representation of the stationary Hawkes process with a non-exponential excitation kernel. From there, by Proposition 12.3.IX of [Reference Daley and Vere-Jones14], we deduce the point process is mixing in the sense of Definition 2.2 below. By using this property together with the notion of mixing in [Reference Clinet and Yoshida7, Section 3.4], we can establish ergodicity of the intensity process. This allows us to identify the ergodic limit to the normalised likelihood function, which, by the M-estimator master theorem [Reference Van der Vaart39, Theorem 5.7], results in consistency of the MLE and is shown herein.
The paper is organised as follows. For the remainder of this section, we formally define the stationary Hawkes process with a general kernel, and some terms and notations that will be used throughout the article. Section 2 is devoted to establishing ergodicity of the intensity process of the stationary Hawkes process. This leads to Section 3, where we consider a parametric framework for and establish consistency of the MLE. We conclude this paper in Section 4. Finally, Appendix A is devoted to the statements and proofs of some boundedness results that are used in the proofs in Sections 2 and 3.
1.1. Setup and notations
This article begins by considering a simple and stationary self-exciting point process N defined on the probability space
$(\Omega, \mathcal{E}, P)$
. Let
$\{\mathcal{F}_t\}_{t \in \mathbb{R}}$
denote the natural filtration induced by N, where
$\mathcal{F}_t = \sigma\{N_s\colon s \in (-\infty, t]\}$
. The intensity process of N relative to
$\{\mathcal{F}_t\}$
is given by
\begin{equation*} \lambda_t = \nu + \int_{-\infty}^{t-} g(t-s) \,\mathrm{d} N_s,\quad t\in \mathbb R. \end{equation*}
It is assumed that
$\nu$
and g satisfy the following conditions.
-
[C1]
-
(i) The baseline intensity satisfies
$0 < \nu < \infty$
. -
(ii) The excitation kernel g(t) is non-negative and satisfies
\begin{align*} 0 < \eta = \int_0^\infty g(s) \,\mathrm{d} s < 1. \end{align*}
-
We can also write
$g = \eta \tilde{g}$
, where
$\tilde{g}$
is the density function of a non-negative random variable. Here
$\nu$
is known as the constant baseline intensity and
$\eta$
is known as the branching ratio.
Let us introduce a few terms and notations. If x is a real number, a vector or a matrix,
$\vert x \vert_p = (\sum_i \vert x_i \vert^p)^{1/p}$
and
$\vert \cdot \vert = \vert \cdot \vert_1$
. If X is a random variable in
$\mathbb{R}^d$
,
$\Vert X \Vert_p = \mathbb{E}[{\sum_i\vert X_i \vert^p}]^{1/p}$
. For two functions
$f_1(\cdot)$
and
$f_2(\cdot)$
,
$f_1(x)={\mathrm{O}} (f_2(x))$
means
$|f_1(x)|\leq M |f_2(x)|$
when
$x\to \infty$
for some finite
$M\geq 0$
. For some Borel space
$(E, \mathcal{B}(E))$
, let
$C_b(E, \mathbb{R})$
denote the set of functions
$\psi\colon E \to \mathbb{R}$
such that
$\psi$
is continuous and bounded, and let
$C_p(E,\mathbb{R})$
denote the set of functions
$\psi\colon E \to \mathbb{R}$
such that
$\psi$
is continuous on E and of polynomial growth of order p in E, that is, for any
$p >0$
,
$\psi (x)= {\mathrm{O}} (\vert x \vert^p)$
for all
$\psi \in C_p(E, \mathbb{R})$
.
2. Main results
Notions of ergodicity, mixing, and stationarity are crucial for studying the dependence structure of stochastic processes. A stationary process is one whose distribution is invariant under shifts, a process that is mixing implies some form of asymptotic independence, and ergodicity encapsulates the idea that certain statistical properties (e.g. state space average) can be deduced through a single and sufficiently long sample (e.g. time average). The study of these properties plays a vital role in developing asymptotic inference theories for stochastic processes. This section begins with a collection of definitions, sourced from the literature, with the goal of communicating the concepts that follow effectively.
A Borel measure
$\mu$
on
$\mathbb{R}$
is boundedly finite if
$\mu(A) <\infty$
for every bounded Borel set A of
$\mathbb{R}$
. Denoting
$\mathcal{M}$
to be the space of all boundedly finite Borel measures on
$\mathbb{R}$
,
$\mathcal{M}$
acquires a natural topology, as well as a family of Borel sets
$\mathcal{B}(\mathcal{M})$
which can be used to define measures on
$\mathcal{M}$
[Reference Daley and Vere-Jones13, page 158]. Thus, a point process
$\xi$
on
$\mathbb{R}$
can be interpreted as a measurable mapping from a probability space
$(\Omega, \mathcal{E}, P)$
into the space
$(\mathcal{M}, \mathcal{B}(\mathcal{M}))$
. Hence, for every sample point
$\omega \in \Omega$
, we associate it with a particular realisation which is a boundedly finite Borel measure on
$\mathbb{R}$
, hereby denoted as
$\xi(\omega, \cdot)$
. Such a realisation has the value
$\xi(\omega, A)$
on the Borel set
$A \in \mathcal{B}(\mathbb{R})$
, and the distribution of a random measure is the probability measure it induces on
$(\mathcal{M}, \mathcal{B}(\mathcal{M}))$
. Note that for each fixed
$A\in \mathcal B(\mathbb R)$
,
$\xi_A :\!= \xi(\cdot, A)$
is a non-negative random variable. Next, we give the following definition of ergodicity of stochastic processes, as presented in [Reference Clinet and Yoshida7].
Definition 2.1. Let
$(E, \mathcal{B}(E))$
be a Borel space and let
$X\colon \Omega \times \mathbb{R}\to E$
be a stochastic process. X is C-ergodic for (or ergodic with respect to) some family of functions
$C\colon E \to \mathbb{R}$
if, for any
$\psi \in C$
, there exists a mapping
$\pi \colon C\to \mathbb{R}$
such that
\begin{equation*} \dfrac{1}{T} \int_{0}^{T} \psi(X_s) \,\mathrm{d} s \overset{\mathbb{P}}{\rightarrow} \pi(\psi) \end{equation*}
as
$T \rightarrow \infty$
. Here,
$\pi(\psi)$
is also known as the ergodic limit of
$\psi(X_s)$
.
This rest of this section is dedicated to proving that the intensity process
$\lambda$
is ergodic, in the sense of Definition 2.1, under certain regularity conditions. We can see from the above definition that ergodicity implies the convergence of the time average of
$\psi(X_s)$
, a stochastic process X under some map
$\psi$
, to some deterministic limit
$\pi(\psi)$
. Once
$\pi(\psi)$
is shown to exist and subsequently identified, analysis can be done on it to infer the asymptotic properties of
$\psi(X_s)$
. This method for asymptotics is particularly convenient when the underlying stochastic process X is non-stationary. We now state the main result concerning the ergodicity of
$\lambda$
.
Theorem 2.1. Under [C1],
$\lambda$
is
$C_b(\mathbb{R}_+,\mathbb{R})$
-ergodic as in Definition 2.1 with
$\pi(\psi) = \mathbb{E}[\psi(\lambda_t)]$
for any
$t \in \mathbb{R}$
.
Before we give the proof of Theorem 2.1, let us introduce the notion of mixing of random measures (or point processes) in general. To do this, let us give a few definitions. The finite-dimensional distributions of a random measure
$\xi$
are the joint distributions, for all finite families of bounded Borel sets
$A_1, \ldots, A_k$
, of the random variables
$\xi(A_1), \ldots, \xi(A_k)$
, which is denoted by
\begin{equation*} P \{ \xi(A_i) \le x_i, i = 1, \ldots, k \}.\end{equation*}
For arbitrary
$u \in \mathbb{R}$
,
$x \in \mathbb{R}$
, and
$A \in \mathcal{B}(\mathbb{R})$
, the translation operator
$T_u$
is defined by
\begin{equation*} T_u x= x + u, \quad T_uA = A+u = \{x + u\colon x \in A\},\end{equation*}
and the shift operator
$S_u$
on
$\mathcal{M}$
is defined by
\begin{equation*} (S_u\mu)(A) = \mu(T_uA), \quad \mu \in \mathcal{M}, A \in \mathcal{B}(\mathbb{R}).\end{equation*}
Note that if
$\xi$
is a random measure then so is
$S_u\xi$
, as it is the composition of two measurable mappings. Finally, a random measure
$\xi$
on
$\mathbb{R}$
is stationary if, for all
$u \in \mathbb{R}$
, the finite-dimensional distributions of
$\xi$
and
$S_u\xi$
coincide; and for some point process
$\xi$
, we let
$\mathcal{P}(\cdot) :\!=a P(\xi \in \cdot)$
denote its distribution on
$(\mathcal{M}, \mathcal{B}(\mathcal{M}))$
. We can now state the following definition.
Definition 2.2.
(Definition 12.3.I. of [Reference Daley and Vere-Jones14].) The stationary random measure
$\xi$
on
$\mathbb{R}$
-
(i) is ergodic if, for all V and
$W \in \mathcal{B}(\mathcal{M})$
,
\begin{equation*} \lim_{u \rightarrow \infty} \dfrac{1}{2u} \int_{-u}^u ( \mathcal{P}(S_x V \cap W) - \mathcal{P}(V) \mathcal{P}(W) ) \,\mathrm{d} x = 0 ; \end{equation*}
-
(ii) is mixing if, for all V and
$W \in \mathcal{B}(\mathcal{M})$
,
\begin{equation*} \lim_{\vert u \vert \rightarrow \infty} \mathcal{P}(V \cap S_u W) - \mathcal{P}(V) \mathcal{P}(W) = 0. \end{equation*}
Definition 2.2 defines the ergodic and mixing properties of random measures in general. While such poperties of random measures are of interest in their own right, as we shall see in the proof of Theorem 2.1, we also need the following definitions of mixing of stochastic processes, as presented in [Reference Clinet and Yoshida7].
Definition 2.3. The stochastic process
$X\colon \Omega \times \mathbb{R} \to E$
is C-mixing for (or mixing with respect to) some family of functions
$C\colon E \to \mathbb{R}$
if, for any
$\phi$
,
$\psi \in C$
,
\begin{equation*} \sup_{t \in \mathbb{R}} \mathrm{cov} (\phi(X_t), \psi(X_{t +u}) ) \rightarrow 0, \quad \vert u \vert \rightarrow \infty. \end{equation*}
Remark 2.1. It is important to note the distinction between Definition 2.2 on ergodicity and the mixing property from Definitions 2.1 and 2.3. Definition 2.2 is formulated in the context of random measures whilst Definitions 2.1 and 2.3 are concerned with stochastic processes.
Proof of Theorem
2.1. In this proof we first establish that the point process N, when considered as a random measure, is mixing in the sense of Definition 2.2. This is achieved by first noting that there exists a stationary Poisson cluster process representation of N, which, in view of Proposition 12.3.IX of [Reference Daley and Vere-Jones14], allows us to deduce Lemma 2.1 below. The next step is to translate the mixing property of N to show that the intensity process
$\lambda$
is mixing with respect to
$C_b(\mathbb{R}_+,\mathbb{R})$
. From here, we adopt the proof of Lemma 3.16 from [Reference Clinet and Yoshida7] to transfer the mixing property of
$\lambda$
to show that
$\lambda$
is
$C_b(\mathbb{R}_+, \mathbb{R})$
-ergodic. These steps are summarised as Lemmas 2.1–2.3 below, the proofs of which can be found in Appendix A.2.
Lemma 2.1. Under [C1], N is mixing in the sense of Definition 2.2.
Lemma 2.2. Under [C1],
$\lambda$
is
$C_b(\mathbb{R}_+,\mathbb{R})$
-mixing.
Lemma 2.3. If
$\lambda$
is
$C_b(\mathbb{R}_+,\mathbb{R})$
-mixing, then
$\lambda$
is
$C_b(\mathbb{R}_+, \mathbb{R})$
-ergodic. Moreover, for any
$\psi \in C_b(\mathbb{R}_+, \mathbb{R})$
and
$t \in \mathbb{R}$
, we have
$\pi(\psi) = \mathbb{E}[\psi(\lambda_t)]$
.
Theorem 2.1 states that
$\lambda$
is
$C_b(\mathbb{R}_+, \mathbb{R})$
-ergodic. However, ergodicity of
$\lambda$
with respect to
$C_b(\mathbb{R}_+,\mathbb{R})$
, the family of continuous and bounded functions from
$\mathbb{R}_+$
to
$\mathbb{R}$
, could limit the theorem’s capacity for application. For example, similarly to [Reference Clinet and Yoshida7] and [Reference Kwan, Chen and Dunsmuir27], if we wish to identify the ergodic limit to, say, the normalised log-likelihood function of a Hawkes process, Theorem 2.1 will not suffice since the mapping of interest does not belong to
$C_b(\mathbb{R}_+, \mathbb{R})$
. For this reason, we require the following proposition, which gives the conditions such that
$\lambda$
is
$C_p(\mathbb{R}_+, \mathbb{R})$
ergodic.
Proposition 2.1. In addition to [C1], for any
$p \ge 1$
, if
$\mathbb{E}[(\lambda_t)^{p+\epsilon}] < \infty$
for some
$\epsilon >0$
, then
$\lambda$
is
$C_{p}(\mathbb{R}_+, \mathbb{R})$
-ergodic. Moreover, for any
$\psi \in C_p(\mathbb{R}_+, \mathbb{R})$
, we have
$\pi(\psi) = \mathbb{E}[\psi(\lambda_t)]$
for any
$t \in \mathbb{R}$
.
Proof. For any
$p \in \mathbb{R}$
with
$p \ge 1$
, suppose
$\psi \in C_p(\mathbb{R}_+, \mathbb{R})$
. Since we can write
$\psi = \psi^+ - \psi^-$
, where both
$\psi^+$
and
$\psi^-$
are non-negative, it is not restrictive to assume
$\psi$
is non-negative. For some
$\delta >0$
, denote
\begin{align*}V_T = \dfrac{1}{T} \int_0^T \psi(\lambda_t) \,\mathrm{d} t\quad\text{and}\quad V^\delta_T =\dfrac{1}{T} \int_0^T \psi(\lambda_t) \land \delta \,\mathrm{d} t.\end{align*}
Let us first show that
$V_T$
converges in probability to some limit, then show that the limit is given by
$\pi(\psi) = \mathbb{E}[\psi(\lambda_t)]$
for any
$t \in \mathbb{R}$
. For some
$\epsilon>0$
, by stationarity of
$\lambda_t$
, we have
\begin{align*} \mathbb{E}\bigl[ \vert V_T - V_T^\delta \vert \bigr] & = \dfrac{1}{T} \mathbb{E}\biggl[ \int_0^T \vert \psi(\lambda_t) - \delta \vert \unicode{x1D7D9}_{ \{\psi(\lambda_t) \ge \delta \} } \,\mathrm{d} t \biggr] \\ &\le \dfrac{1}{T} \mathbb{E}\biggl[ \int_0^T \psi(\lambda_t) \unicode{x1D7D9}_{ \{\psi(\lambda_t) \ge \delta \} } \,\mathrm{d} t \biggr] \\ & \le \dfrac{1}{T \delta^{\epsilon}} \mathbb{E}\biggl[ \int_0^T \psi(\lambda_t) ^{1+\epsilon} \,\mathrm{d} t \biggr] \\ &= \dfrac{1}{\delta^\epsilon} \mathbb{E}\bigl[ \psi(\lambda_t)^{1+\epsilon} \bigr]. \end{align*}
Since
$\psi$
is of polynomial growth of order p and
$\epsilon$
is arbitrary, thus, if
$ \mathbb{E}[(\lambda_t)^{p(1+{\epsilon} ) }] < \infty$
, then
$ \mathbb{E}[ \psi(\lambda_t)^{1+\epsilon} ]$
is bounded by some finite constant
$K>0$
. Let T and
$T' \in \mathbb{R}_+$
; then
\begin{align*} \mathbb{E}\bigl[ \vert V_T - V_{T'} \vert \bigr] \le \mathbb{E}\bigl[ \vert V_T - V_T^\delta \vert \bigr] + \mathbb{E}\bigl[ \vert V_{T'} - V_{T'}^\delta \vert \bigr] + \mathbb{E}\bigl[ \vert V_T^\delta - V_{T'}^\delta \vert \bigr] \le \dfrac{2K}{\delta^\epsilon} + \mathbb{E}\bigl[ \vert V_T^\delta - V_{T'}^\delta \vert \bigr] . \end{align*}
Since
$\psi \land \delta \in C_b(\mathbb{R}_+, \mathbb{R})$
and, by hypothesis,
$\mathbb{E}[(\lambda_t)^{p+\epsilon}] < \infty$
for some
$\epsilon > 0$
,
$V^\delta_T$
is bounded in
$\mathbb{L}_1$
. By Theorem 2.1 and the
$\mathbb{L}_p$
-convergence theorem [Reference Kallenberg24, Proposition 5.12],
$V^\delta_T \rightarrow \pi(\psi \land \delta)$
in
$\mathbb{L}_1$
and therefore
\begin{equation*} \mathbb{E}\bigl[ \vert V^\delta_T - V^\delta_{T'} \vert \bigr] \le \mathbb{E}\bigl[ \vert V^\delta_T -\pi(\psi \land \delta) \vert \bigr] + \mathbb{E}\bigl[ \vert V^\delta_{T'} -\pi(\psi \land \delta) \vert \bigr] \end{equation*}
can be made arbitrarily small by controlling T and T’. Thus far, we have shown that for any
$\epsilon >0$
, by taking
$\delta$
, T, and T’ to be sufficiently large, we have
$\mathbb{E}[\vert V_T - V_{T'}\vert] \le \epsilon$
. This shows that
$V_T$
is a Cauchy sequence in T and hence converges to some limit in probability. It follows readily from the proof of Proposition 3.8 in [Reference Clinet and Yoshida7] that the in-probability limit is given by
$\pi(\psi) = \mathbb{E}[\psi(\lambda_t)]$
for any
$t \in \mathbb{R}$
.
3. Application to parametric Hawkes processes
In this section we introduce the parametric Hawkes process and make extensive use of the ergodic theories developed in Section 2 to establish consistency of the MLE. This section consists of two subsections. Section 3.1 introduces the stationary parametric Hawkes process and is devoted to translating the ergodic results from Section 2 to a parametric context, whereas in Section 3.2 we give sufficient conditions such that the MLE is a consistent estimator.
3.1. Stationary parametric Hawkes process
Suppose there exists a finite-dimensional parameter
$\theta^0$
from some compact parameter space
$\Theta \subset \mathbb{R}^d$
such that given
$\theta^0$
, the true intensity of
$N_t$
relative to
$\mathcal{F}_t$
, which is denoted by
$\lambda_{t, \theta^0}$
, is known. For any
$\theta \in \Theta$
, we define
\begin{equation} \lambda_{t,\theta} = \nu(\theta) + \int_{-\infty}^{t-} g(t-s;\,\theta ) \,\mathrm{d} N_s.\end{equation}
Note that there exists only one stationary and orderly point process whose complete intensity satisfies (3.1) [Reference Hawkes and Oakes21]. To study the limiting behaviour of the compensated central log-likelihood function, we need to consider the bivariate process
$Y_{t, \theta, \theta^0} = (\lambda_{t, \theta}, \lambda_{t, \theta^0}) $
. Letting
$E_1 = \mathbb{R}_+ \times \mathbb{R}_+$
, we shall show that, under suitable regularity conditions,
$Y_{t,\theta, \theta^0}$
is
$C_p(E_1, \mathbb{R})$
-ergodic. This is done by first showing that
$Y_{t, \theta, \theta^0}$
is
$C_b(E_1, \mathbb{R})$
-ergodic and then generalising it to be
$C_p(E_1,\mathbb{R})$
-ergodic using an argument similar to Proposition 2.1. We now give the first set of regularity conditions.
-
[C2]
-
(i) The excitation kernel
$g(t;\,\theta)$
is non-negative and satisfies where
\begin{align*} g(t;\,\theta) \equiv \eta (\theta) \tilde{g}(t;\,\theta), \end{align*}
$\tilde{g}$
is the proper density function of a positive random variable that depends only on
$\theta$
.
-
(ii)
$\nu$
, g, and
$\eta$
depend on some finite-dimensional parameter
$\theta=(\theta_1, \ldots, \theta_d)^\top$
that belongs to some parameter space
$\Theta \subset\mathbb{R}^{d}$
. Furthermore, we assume that
$\Theta$
is compact and contains a non-empty open ball in
$\mathbb{R}^d$
which, in turn, contains
$\theta^0$
, with
$\theta^0$
being the true parameter. -
(iii) There exist
$\underline{\nu}$
,
$\bar{\nu}$
,
$\underline{\eta}$
,
$\bar{\eta} \in \mathbb{R}_+$
such that
$0 < \underline{\nu} \le \nu(\theta) \le \bar{\nu} < \infty$
and
$0< \underline{\eta} \le \eta(\theta) \le \bar{\eta}< 1$
for all
$\theta \in \Theta$
.
-
The uniform bounds stated in [C2](iii) are used to establish boundedness of the intensity process in
$\mathbb{L}_p$
, as shown in Lemma A.2. We now state the ergodic theorem concerning
$Y_{t, \theta, \theta^0}$
, which is analogous to Theorem 2.1.
Theorem 3.1. Under [C2],
$Y_{t, \theta, \theta^0}$
is
$C_b(E_1, \mathbb{R})$
-ergodic with
$\pi(\theta, \theta^0;\,\psi) = \mathbb{E}[\psi(Y_{t, \theta, \theta^0})]$
for any
$t \in \mathbb{R}$
. That is, there exists a mapping
$\pi\colon \Theta \times \Theta \times C_b(E_1,\mathbb{R}) \to \mathbb{R}$
such that for any
$\theta, \theta^0 \in \Theta$
,
$\psi \in C_b(E_1, \mathbb{R})$
, we have
\begin{equation*} \biggl\vert \dfrac{1}{T} \int_{0}^{T} \psi\bigl(Y_{t,\theta, \theta^0} \bigr)\,\mathrm{d} t - \pi\bigl(\theta, \theta^0;\, \psi\bigr) \biggr\vert \overset{\mathbb{P}}{\rightarrow} 0, \quad T \rightarrow \infty. \end{equation*}
Proof of Theorem 3.1. By Lemma 2.1, N is mixing. Thus, similarly to the proof of Theorem 2.1, it remains to first show that
$Y_{t, \theta, \theta^0}$
is
$C_b(E_1, \mathbb{R})$
-mixing from the mixing property of N and, from there, establish
$C_b(E_1, \mathbb{R})$
-ergodicity of
$Y_{t, \theta, \theta^0}$
. In fact, this proof is no different from the proof of Theorem 2.1. To see this, by fixing
$\theta$
and
$\theta^0$
, note that the regularity conditions imposed on
$\nu$
and g under [C2] ensure that both
$\lambda_{t, \theta}$
and
$\lambda_{t, \theta^0}$
can be regarded as measurable functions that map
$\mathcal{B}(\mathcal{N}) \times \mathbb{R}$
to
$\mathbb{R}_+$
. Hence the proof to show that
$Y_{t, \theta, \theta^0}$
is
$C_b(E_1,\mathbb{R})$
-mixing follows readily from the proof of Lemma 2.2. Similarly, it follows immediately from the proof of Lemma 2.3 that
$Y_{t, \theta, \theta^0}$
is
$C_b(E_1, \mathbb{R})$
-ergodic.
To extend the above ergodic result to the class of functions
$C_p(E, \mathbb{R})$
, we require additional conditions on the moment of
$\lambda_{t, \theta}$
.
Proposition 3.1. In addition to [C2], for any
$p \ge 1$
, if
$\mathbb{E}[(\lambda_{t, \theta})^{p + \epsilon}]< \infty$
for some
$\epsilon >0$
, then
$Y_{t, \theta, \theta^0}$
is
$C_p(E_1, \mathbb{R})$
-ergodic.
Proof of Proposition 3.1. This proof is a direct adaptation of the proof of Proposition 2.1 and the details are omitted.
3.2. Consistency
The discussion in the previous subsection is concerned with the stationary Hawkes process N, which is a point process defined on
$\mathbb{R}$
. However, in practice, only events starting from time 0 are observable. For this reason, let us consider the non-stationary version of N which is denoted by
$\tilde{N}$
observed on the interval [0, n]. Let
$\tilde{\mathcal{F}}_t = \sigma\{\tilde{N}_s\colon 0 \le s \le t\}$
denote the
$\sigma$
-algebra generated by
$\tilde{N}$
, and let
$\tilde{\mathcal{F}} = (\tilde{\mathcal{F}}_t)_{t \in [0,n]}$
denote the natural filtration to which
$\tilde{N}$
is adapted. The intensity of
$\tilde{N}_t$
relative to
$\tilde{\mathcal{F}}_t$
evaluated at
$\theta$
is
\begin{equation*} \tilde{\lambda}_{t, \theta} = \nu(\theta) + \int_0^{t-} g(t-s;\, \theta) \,\mathrm{d} \tilde{N}_s.\end{equation*}
The normalised log-likelihood function corresponding to some realisation of
$\tilde{N}$
on the interval [0, n], up to an additive constant, is
\begin{equation*} \tilde{\mathcal{L}}^n(\theta) = \dfrac{1}{n} \int_0^n \log(\tilde{\lambda}_{t, \theta}) \,\mathrm{d} \tilde{N}_t - \dfrac{1}{n}\int_0^n \tilde{\lambda}_{t, \theta} \,\mathrm{d} t.\end{equation*}
The MLE
$\hat{\theta}^n$
maximises the log-likelihood
$\tilde{\mathcal{L}}^n(\theta)$
over
$\Theta$
. Next, we give sufficient conditions, in addition to [C2], to establish consistency of the MLE.
-
[C3]
-
(i) For any
$\theta \in \Theta$
,
$g(t;\,\theta)^2$
is integrable. Furthermore, the functions
$\nu(\theta)$
and
$\eta(\theta)$
are equicontinuous, and
$g(t;\, \theta)$
is uniformly equicontinuous when regarded as a family of functions of
$\theta$
. -
(ii) For any
$\theta$
,
$\theta' \in \Theta$
and
$t\in \mathbb{R}_+$
,
$\tilde{\lambda}_{t, \theta} = \tilde{\lambda}_{t, \theta'}$
almost surely only when
$\theta = \theta'$
.
-
Remark 3.1. It can be shown that the generalised Pareto, Gamma, and Weibull families of density functions satisfy the conditions imposed on the excitation kernel under [C2]. The square integrability condition under [C3](i) requires the shape parameter for the Gamma and Weibull kernel to be greater than
$\tfrac{1}{2}$
. Compared to the exponential kernel, the generalised Pareto kernel has a heavier tail since it decays toward 0 at a polynomial rate. Therefore, when modelling phenomena where events in the past have a substantially longer lasting effect on the current intensity, the generalised Pareto kernel may be the preferred candidate. On the other hand, since both the exponential and generalised Pareto kernels decay monotonically, if we wish to model events where the peak excitation effect of an occurrence is not immediate, but rather at a delay, the Gamma or Weibull kernel with shape parameter greater than 1 may provide a better fit.
We now state the theorem on the consistency of the MLE.
Theorem 3.2. (Consistency.) Under [C2] and [C3], with probability tending to 1,
$\hat{\theta}^n$
exists as a maximiser to the normalised log-likelihood function
$\tilde{\mathcal{L}}^n(\theta)$
and
$\hat{\theta}^n \overset{\mathbb{P}}{\rightarrow} \theta^0$
.
Proof of Theorem 3.2. Similarly to the proof of Theorem 3 from [Reference Kwan, Chen and Dunsmuir27], in this proof, we will make use of the M-estimator master theorem [Reference Van der Vaart39, Theorem 5.7] to establish consistency of
$\hat{\theta}^n$
. This is done by sequentially showing the following.
-
(i) There exists some deterministic function
$\pi\colon \Theta \times \Theta \to \mathbb{R}$
such that (3.2)
\begin{equation} \sup_{\theta\in\Theta} \vert \tilde{\mathcal{L}}^n(\theta) - \tilde{\mathcal{L}}^n(\theta^0) - \pi(\theta, \theta^0) \vert \overset{\mathbb{P}}{\rightarrow} 0. \end{equation}
-
(ii)
$\theta^0$
exists as a well-separated point of maximum to
$\pi(\theta, \theta^0)$
; that is, for any
$\epsilon >0$
, there exists an open ball
$B(\theta^0, \epsilon)$
centred at
$\theta^0$
with radius
$\epsilon$
such that
$\sup_{\theta \in \Theta \backslash B(\theta^0, \epsilon)} \pi(\theta, \theta^0) < \pi(\theta^0, \theta^0)$
.
To see that (3.2) holds, we first show that
$\tilde{\mathcal{L}}^n(\theta) - \tilde{\mathcal{L}}^n( \theta^0) - \pi(\theta, \theta^0)$
converges to 0 in probability, and then generalise the convergence to be uniform in
$\theta$
. Note that we can bound
$\vert \tilde{\mathcal{L}}^n(\theta) - \tilde{\mathcal{L}}^n(\theta^0) - \pi(\theta, \theta^0) \vert$
by
\begin{align} & \vert \tilde{M}^n(\theta, \theta^0) \vert + \vert \tilde{L}^n(\theta, \theta^0) - \tilde{L}^n(\theta^0, \theta^0) - L^n(\theta, \theta^0) + L^n(\theta^0, \theta^0) \vert \nonumber \\ &\quad + \vert L^n(\theta, \theta^0) - L^n(\theta^0, \theta^0) - \pi(\theta, \theta^0) \vert , \end{align}
where
\begin{align*}\tilde{M}^n(\theta, \theta^0) = \dfrac{1}{n} \int_{0}^{n} \log(\tilde{\lambda}_{t,\theta}/\tilde{\lambda}_{t,\theta^0}) \,\mathrm{d} \bar{\tilde{N}}_t\end{align*}
with
\begin{align*}\bar{\tilde{N}}_t = \tilde{N}_t - \int_0^t \tilde{\lambda}_{s,\theta^0}\,\mathrm{d} s\end{align*}
denoting the compensated process,
\begin{align*} \tilde{L}^n(\theta, \theta^0) = \dfrac{1}{n}\int_0^n \log(\tilde{\lambda}_{t,\theta}) \tilde{\lambda}_{t,\theta^0} - \tilde{\lambda}_{t,\theta} \,\mathrm{d} t, \end{align*}
and
\begin{align*}L^n(\theta, \theta^0) = \dfrac{1}{n}\int_0^n \log(\lambda_{t,\theta}) \lambda_{t,\theta^0} - \lambda_{t,\theta} \,\mathrm{d} t.\end{align*}
By Proposition 3.1, the last term of (3.3) converges to 0 in probability. Next, we turn to showing that the first two terms of (3.3) also converge to 0 in probability. This leads to the following lemmas, the proofs of which can be found in Appendix A.3.
Lemma 3.1. Under [C2] and [C3],
$\tilde{M}^n(\theta, \theta^0) \to 0$
in probability.
Lemma 3.2. Under [C2] and [C3], for any
$\theta \in \Theta$
,
$\Vert \tilde{L}^n(\theta, \theta^0) - L^n(\theta, \theta^0) \Vert \to 0$
.
In view of Lemma 3.1, Lemma 3.2, and the above discussion, we deduce that under [C2] and [C3],
$\vert \tilde{\mathcal{L}}^n(\theta) - \tilde{\mathcal{L}}^n(\theta^0)- \pi(\theta, \theta^0) \vert \to 0$
in probability. The next step is to show that this convergence is uniform in
$\theta$
. While assuming that [C2] and [C3] hold, note that for any
$\epsilon >0$
, there exists
$\delta >0$
such that for all
$\theta$
,
$\theta' \in \Theta$
with
$\vert \theta - \theta' \vert < \delta$
, we have
$\vert \tilde{\mathcal{L}}^n(\theta) - \tilde{\mathcal{L}}^n(\theta')\vert \le \epsilon / 3$
with probability tending to 1. In addition, since
$\nu(\theta)$
,
$\eta(\theta)$
, and
$\tilde{g}(t;\, \theta)$
are uniformly equicontinuous [Reference Kolmogorov and Fomin25] when regarded as families of functions of
$\theta$
, and
$\pi(\theta, \theta^0) = \mathbb{E}[\psi(Y_{t,\theta,\theta^0})]$
for some
$\psi \in C_p(E_1,\mathbb{R})$
, it can be shown that for the same
$\epsilon$
,
$\theta$
,
$\theta'$
, and
$\delta$
,
$\vert \pi(\theta, \theta^0) - \pi(\theta', \theta^0) \vert \le\epsilon/3$
. Furthermore,
$\Theta$
is a compact set in
$\mathbb{R}^d$
, so there exists a finite number of open balls with radius
$\delta$
whose union covers
$\Theta$
. Let
$\tilde{\theta}_1, \ldots, \tilde{\theta}_k$
be the centre of these balls, and for all
$\theta$
, let
$\tilde{\theta}_{i(\theta)}$
be the centre of a ball containing
$\theta$
; then
$\mathbb{P} ({\sup}_{\theta \in \Theta} \vert \tilde{\mathcal{L}}^n(\theta) -\tilde{\mathcal{L}}^n(\theta^0) - \pi(\theta, \theta^0) \vert >\epsilon) $
is bounded by
\begin{align*} & \mathbb{P} \Big( \sup_{\theta \in \Theta} \vert \tilde{\mathcal{L}}^n(\theta) - \tilde{\mathcal{L}}^n(\tilde{\theta}_{i(\theta)}) \vert > {\epsilon}/{3} \Big) + \mathbb{P} \Big(\sup_{\theta \in \Theta} \vert \tilde{\mathcal{L}}^n(\tilde{\theta}_{i(\theta)}) - \tilde{\mathcal{L}}^n(\theta^0) - \pi(\tilde{\theta}_{i(\theta)}, \theta^0) \vert > \epsilon/3\Big) \\&\quad + \mathbb{P} \Big( \sup_{\theta \in \Theta} \vert \pi(\tilde{\theta}_{i(\theta)}), \theta^0\Big) - \pi(\theta, \theta^0) \vert > \epsilon/3 ).\end{align*}
By definition
$\vert \tilde{\theta}_{i(\theta)} - \theta \vert \le \delta$
, hence the first term above converges to 0 as n tends to infinity and the last term equals 0. Furthermore, the second term above is bounded by
\begin{equation*} \sum_{i=1}^k \mathbb{P} \biggl( \vert \tilde{\mathcal{L}}^n(\tilde{\theta}_{i(\theta)}) - \tilde{\mathcal{L}}^n(\theta^0) - \pi(\tilde{\theta}_{i(\theta)}, \theta^0) \vert > \dfrac{\epsilon}{3}\biggr),\end{equation*}
which converges to 0 as
$n \to \infty$
. This shows (3.2).
To complete the proof, it remains to show that
$\theta^0$
is a suitably separated point of maximum of
$\pi(\theta,\theta^0)$
. By Proposition 3.1, we have
\begin{equation} \pi (\theta, \theta^0 ) = \mathbb{E} \bigl[ \log (\lambda_{t, \theta} / \lambda_{t, \theta^0} ) \lambda_{t, \theta^0} - \lambda_{t, \theta} + \lambda_{t, \theta^0} \bigr].\end{equation}
Note that
$\log(x/y)y - x + y \le 0$
for x,
$y > 0$
with
$\log(x/y)y-x+y =0$
if and only if
$x = y$
. We deduce
$\pi(\theta, \theta^0) \le 0$
for all
$\theta \in \Theta$
. Since
$\pi(\theta^0, \theta^0) = 0$
, the next lemma ensures
$\theta^0$
exists as a well-separated point of maximum to
$\pi(\theta, \theta^0)$
, the proof of which is given in Section A.2.
Lemma 3.3. Under [C2] and [C3], for all
$\epsilon >0$
, there exists an open ball
$B(\theta^0, \epsilon)$
centred at
$\theta^0$
with radius
$\epsilon$
such that
\begin{align*} \sup\limits_{\theta \in \Theta \backslash B(\theta^0, \epsilon)} \pi (\theta, \theta^0 ) < 0.\end{align*}
Remark 3.2. It is not necessary for
$g^2$
to be integrable in order for the last term of (3.3) to converge to 0 in probability. Instead, we can replace the integrability condition with a moment-type condition on
$\lambda_{t, \theta}$
. In particular, for any arbitrarily small
$\epsilon > 0$
, the integrand of
$L^n(\theta, \theta^0)$
can be bounded by functions of polynomial growth in
$Y_{t, \theta, \theta^0}$
of order
$1 +\epsilon$
. Therefore, in view of Proposition 3.1, boundedness of
$\Vert \lambda_{t,\theta}\Vert^{1+\epsilon}$
is sufficient for
$ L^n(\theta, \theta^0) - L^n(\theta^0, \theta^0) - \pi(\theta, \theta^0) \rightarrow 0$
in probability.
Theorem 3.2 states sufficient conditions such that the MLE is consistent. To establish asymptotic normality of the MLE, we need to consider the score function and Hessian matrix, which are expressions based on the first and second derivatives of
$\lambda$
. Hence, to adopt the previous procedure, the ergodic properties established thus far need to be extended to the joint ergodicity of
$(\lambda_{t, \theta}, \lambda_{t, \theta^0}, \partial_\theta\lambda_{t, \theta}, \partial_\theta^{\otimes 2} \lambda_{t, \theta})$
, where
$\partial_\theta^{\otimes 2}$
denotes the Kronecker product of
$\partial_\theta$
and its transpose, that is,
$\partial_\theta^{\otimes 2} = \partial_\theta{\partial_\theta}^\intercal \in \mathbb{R}^{d \times d}$
. To do this, additional smoothness conditions on the excitation kernel and its derivatives are required. For the Pareto, Gamma, and Weibull families of kernels mentioned previously, these additional smoothness conditions can be shown to hold under suitable restrictions on the parameters, the details of which can be found in [Reference Kwan26, Chapter 4].
4. Conclusion
This article is dedicated to investigating the asymptotic properties of a stationary Hawkes process where the baseline intensity is constant and the excitation kernel is non-exponential. By considering the stationary Hawkes process as a Poisson cluster process, we are able to show that the intensity process is ergodic. Since the excitation kernel is non-exponential, a considerable amount of care is required to keep the conditions imposed on the model to a minimum.
Since in practice only events starting from time 0 are observed, we also investigated the ergodic properties of the non-stationary Hawkes process with a constant baseline intensity and a non-exponential kernel in a parametric context. As we have shown, the intensity process is also ergodic. Furthermore, by taking a likelihood approach, we established consistency of the MLE.
While the assumption of a constant baseline intensity limits the modelling capacity of the process, a non-exponential excitation kernel offers substantial benefits through its ability to model phenomena whose peak excitation effect from an event is not immediate and does not decay towards 0 exponentially. In [Reference Kwan, Chen and Dunsmuir27], an approximation procedure is employed to establish asymptotic ergodicity of the intensity of a Hawkes process with a time-varying baseline intensity and an exponential excitation kernel. With moderate modifications in the technical arguments presented in [Reference Kwan, Chen and Dunsmuir27] and additional smoothness-type regularity conditions, ergodicity of the intensity of a Hawkes process with time-varying baseline intensity and non-exponential excitation kernel is achievable, the details of which are available in [Reference Kwan26].
Let us conclude this work by pointing out the crucial role of ergodicity in the development of asymptotic inference theories for stochastic processes in general. The current literature suggests that ergodicity of the Hawkes process is often assumed [Reference Ogata32] or is a consequence of the intensity process being Markovian by assuming an exponential excitation kernel [Reference Clinet and Yoshida7, Reference Kwan, Chen and Dunsmuir27]. When the excitation kernel is non-exponential, it is substantially more difficult to establish ergodicity since the resulting intensity process is non-Markovian, therefore rendering standard techniques for asymptotic inference of Markov processes futile. By successfully establishing ergodicity of the intensity process for the stationary Hawkes process with a non-exponential kernel and constant baseline intensity, this work offers valuable insights into the theoretical properties of Hawkes processes and their extensions, such as Hawkes processes with time-varying baseline intensity and non-exponential kernels, and renewal Hawkes processes.
Appendix A. Technical lemmas and proofs of main results
This appendix consists of two sections : the first section gives the statements and their proofs of some preliminary results which are used throughout this article, and the second section gives the proofs of the lemmas and propositions in Sections 2 and 3. In the arguments presented in this Appendix, K denotes some finite and positive constant, and
$K_p$
denotes some finite and positive constant that depends only on p, both of which might change from line to line, since we are only interested in their bounding qualities.
A.1. Preliminary results
In Proposition 2.1 we require
$\mathbb{E}[(\lambda_t)^{p+\epsilon}]$
to be bounded in addition to [C1] in order for
$\lambda_t$
to be
$C_p(\mathbb{R}_+,\mathbb{R})$
-ergodic. However, moment-type conditions on the intensity process may not be straightforward to verify. For this reason, in this section we develop some preliminary results concerning the conditions on the model setup which would guarantee the boundedness of the pth moment of
$\lambda_t$
,
$\lambda_{t, \theta}$
, and
$\tilde{\lambda}_{t, \theta}$
.
Lemma A.1. In addition to [C1], if
$g^p$
is integrable for some
$p \in \mathbb{N}$
, then
$\sup_{t \in \mathbb{R}} \Vert \lambda_t \Vert_p < \infty$
.
Proof. We will prove by induction that if
$g^p$
is integrable, then
$\sup_{t \in \mathbb{R}} \Vert \lambda_t \Vert_p < \infty$
. Note that [C1] implies g is integrable. Hence it suffices to show that
$\sup_{t \in \mathbb{R}}\Vert \lambda_t \Vert < \infty$
under [C1] for the initialising case. Note that
$\lambda_t \ge 0$
for all t, hence we have
$\Vert \lambda_t \Vert= \mathbb{E}[\lambda_t]$
. By Fubini’s theorem, we can write
\begin{align*} \Vert \lambda_t \Vert= {} \nu + \int_{-\infty}^{t-} g(t-s) \mathbb{E} [\lambda_s ] \,\mathrm{d} s \le {} \nu + \eta \sup_{t \in \mathbb{R}} \mathbb{E} [ \lambda_t]. \end{align*}
By taking the supremum on both sides of the above expression with respect to t and rearranging the terms, it is straightforward to see
$\sup_{t \in \mathbb{R}} \Vert \lambda_t \Vert \le \nu/(1-\eta) < \infty$
.
Next, we will show that the statement holds for any
$p \in \mathbb{N}$
. First, note that for
$p \ge 1$
we have
$ (x + y)^{p} \le (1+\epsilon)^{p-1} x^{p} + (1+\epsilon^{-1})^{p-1}y^{p}$
for all
$x,y,\epsilon >0$
. To see this, write
\begin{align*}(x+y)^p = \biggl(\dfrac{1}{1+\epsilon}(1+\epsilon)x + \dfrac{1}{1+\epsilon^{-1}}(1+\epsilon^{-1})y\biggr)^p\end{align*}
and use the convexity of the function
$(\cdot)^p$
on
$[0,\infty)$
. By the aforementioned inequality,
$\mathbb{E}[ ( \lambda_t )^{p}]$
is bounded by
\begin{equation} (1+\epsilon)^{p-1}\nu^{p} + (1+\epsilon^{-1})^{p-1} \mathbb{E} \biggl[\biggl(\int_{-\infty}^{t-} g(t-s) \,\mathrm{d} N_s \biggr)^{p}\biggr]. \end{equation}
The first term of (A.1) is bounded under [C1]. To see that the second term is also bounded, let us denote the compensated point process by
$\bar{N}_t:\!=a \mathrm{d} N_t - \lambda_t \,\mathrm{d} t$
. Then, similarly to (A.1), for any
$\epsilon >0$
,
\begin{align*}\mathbb{E}\biggl[\biggl( \int_{-\infty}^{t-} g(t-s) \,\mathrm{d} N_s\biggr)^p\biggr]\end{align*}
is bounded by
\begin{align*} (1+\epsilon)^{p-1} \mathbb{E} \biggl[\biggr\vert\int_{-\infty}^{t-} g(t-s) \,\mathrm{d} \bar{N}_s \biggl\vert^{p}\biggr] + (1+\epsilon^{-1})^{p-1} \mathbb{E} \biggl[\biggl(\int_{-\infty}^{t-} g(t-s) \lambda_s \,\mathrm{d} s \biggr)^{p}\biggr] .\end{align*}
To bound the martingale term
\begin{align*}\mathbb{E} \biggl[\biggl\vert\int_{-\infty}^{t-} g(t-s) \,\mathrm{d} \bar{N}_s\biggr\vert^{p}\biggr]\end{align*}
from above, we wish to call on Lemma A.2 of [Reference Clinet and Yoshida7]. However, Lemma A.2 of [Reference Clinet and Yoshida7] is not directly applicable here since N is defined on
$\mathbb{R}$
and the limit of the integral goes from
$-\infty$
to
$t-$
instead of 0 to
$t-$
. To overcome this, we can centre N so that it starts at 0 (i.e.
$N_0 = 0 $
), and write
\begin{equation*} N_t = \begin{cases} N^+_t :\!=\,N( (0, t] ) & t > 0, \\ 0 = N^+_0 = N^-_0 & t = 0, \\ N^-_{-t}\, :\!= - N( [t,0) ) & t < 0. \end{cases}\end{equation*}
$N_t$
,
$N^+_t$
, and
$N^-_t$
can be viewed as random variables indexed by t, and N(A) as a random measure that counts the number of points in A for any
$A \in \mathcal{B}(R)$
(for general discussions of point processes defined on the whole line, see Chapter 3 of [Reference Daley and Vere-Jones13]). A short calculation shows that the compensated processes
$\bar{N}^+_t :\!= N^+_t - \int_0^t \lambda_s \,\mathrm{d} s$
and
$\bar{N}^-_t :\!= N^-_t + \int_{-t}^0 \lambda_s \,\mathrm{d} s$
are both local martingales on
$\mathbb{R}_+$
with
$\bar{N}^+_0 = \bar{N}^-_0 = 0$
. In view of this,
\begin{align*}\mathbb{E}\biggl[ \biggl\vert \int_{-\infty}^{t-} g(t-s) \,\mathrm{d} \bar{N}_s \biggr\vert^p\biggr]\end{align*}
is bounded by
\begin{align*} & ( 1 + \epsilon )^{p-1} \mathbb{E} \biggl[ \biggr\vert \int_{-\infty}^0 g(t-s) \,\mathrm{d} \bar{N}_s \biggl\vert^p \biggr] + ( 1 + \epsilon^{-1} )^{p-1} \mathbb{E} \biggl[ \biggr\vert \int_0^t g(t-s) \,\mathrm{d} \bar{N}_s \biggl\vert^p \biggr] \\ & \quad = (1+\epsilon )^{p-1} \mathbb{E} \biggl[ \biggr\vert \int_0^\infty g(t+s) \,\mathrm{d} \bar{N}^-_s \biggl\vert^p \biggr] + ( 1 + \epsilon^{-1} )^{p-1} \mathbb{E} \biggl[ \biggr\vert \int_0^t g(t-s) \,\mathrm{d} \bar{N}^+_s \biggl\vert^p \biggr].\end{align*}
Note that Lemma A.2 of [Reference Clinet and Yoshida7] is applicable, and for
\begin{align*}E \biggl[\biggl\vert\int_0^\infty g(t+s) \,\mathrm{d} \bar{N}^-_s \biggr\vert^{p}\biggr]\end{align*}
it yields the following bound:
\begin{align} K_p \mathbb{E} \biggl[\int_0^\infty g(t+s)^{p} \lambda_{-s} \,\mathrm{d} s \biggr] + K_p \mathbb{E} \biggl[ \biggl( \int_0^\infty g(t+s)^{2} \lambda_{-s} \,\mathrm{d} s \biggr)^{{p}/{2}} \biggr] . \end{align}
By the assumed integrability of
$g^p$
and the boundedness of
$\sup_{t \in \mathbb{R}} \mathbb{E}[\lambda_t]$
, we deduce that the first term of (A.2) is bounded. Note that by Hölder’s inequality we have
\begin{align*}\biggl(\int f h\biggr)^i \le \biggl(\int f^i h\biggr)\biggl(\int h\biggr)^{i-1}\end{align*}
for
$i\ge 1$
and non-negative functions f and h. Thus, for
$i \ge 1$
and
$j \ge 1$
,
\begin{align*}\mathbb{E}\biggl[\biggl(\int_0^\infty g(t+s)^i \lambda_{-s} \,\mathrm{d} s\biggr)^j\biggr]\end{align*}
is bounded by
\begin{align}& \mathbb{E} \biggl[ \biggl( \int_0^\infty g(t+s)^i (\lambda_{-s} )^j \,\mathrm{d} s \biggr) \biggl( \int_0^\infty g(t+s)^i \,\mathrm{d} s \biggr)^{j-1} \biggr] \notag \\&\quad \le \sup_{t \in \mathbb{R}} \mathbb{E} [(\lambda_t )^{j} ] \biggl(\int_0^\infty g(t+s)^i \,\mathrm{d} s \biggr)^{j}. \end{align}
Furthermore, in addition to [C1], if
$g^p$
is integrable for some
$p >1$
, then for any
$q \in [1, p]$
,
$g^q$
is also integrable since
\begin{align*} \int g^q = \int g^q \unicode{x1D7D9}_{\{ g \le 1 \} } + \int g^q \unicode{x1D7D9}_{\{g > 1 \}} \le \int g \unicode{x1D7D9}_{\{ g \le 1 \} } + \int g^p \unicode{x1D7D9}_{\{g > 1 \}} \le \int g + \int g^p < \infty. \end{align*}
In view of the above discussion and the induction hypothesis, we deduce that the second term of (A.2) is bounded and therefore so is
\begin{align*}\mathbb{E} \biggl[\biggl\vert\int_{-\infty}^0 g(t-s) \,\mathrm{d} \bar{N}_s\biggr\vert^{p}\biggr].\end{align*}
To get a similar estimate for
\begin{align*}\mathbb{E} \biggl[\biggl\vert\int_0^{t-} g(t-s) \,\mathrm{d} \bar{N}_s \biggr\vert^{p}\biggr],\end{align*}
by Lemma A.2 of [Reference Clinet and Yoshida7] again, it is bounded by
\begin{align*} K_p \mathbb{E} \biggl[\int_0^{t-} g(t-s)^{p} \lambda_{s} \,\mathrm{d} s \biggr] + K_p \mathbb{E} \biggl[ \biggl( \int_0^{t-} g(t-s)^{2} \lambda_{s} \,\mathrm{d} s \biggr)^{{p}/{2}} \biggr] .\end{align*}
The arguments used to show that the above expression is bounded are analogous to the ones used to show that (A.2) is bounded, and the details are omitted here. This shows that
\begin{align*}\mathbb{E}\biggl[ \biggl\vert \int_{-\infty}^{t-} g(t-s) \,\mathrm{d} \bar{N}_s \biggr\vert^p\biggr]\end{align*}
is bounded. Finally, similarly to (A.3), by Hölder’s inequality,
\begin{align*}\mathbb{E}\biggl[\biggl(\int_{-\infty}^{t-} g(t-s) \lambda_s \,\mathrm{d} s \biggr)^p\biggr]\end{align*}
is bounded by
$\eta^p \sup_{t \in \mathbb{R}} [\lambda_t]^p$
. Collecting the above bounds, we have shown that
\begin{equation*} \mathbb{E}[(\lambda_t )^p ] \le (1+\epsilon)^{p-1} \nu^p + ( (1+\epsilon) (1+\epsilon^{-1}))^{p-1} K_p + (1+\epsilon^{-1})^{2p-2} \eta^p \sup_{t \in \mathbb{R}} \mathbb{E} [(\lambda_t )^p ].\end{equation*}
By letting
$\epsilon$
be sufficiently large that
$(1+\epsilon^{-1})^{2p-2} \eta^p < 1$
, taking the supremum on both sides of the above expression in t over
$\mathbb{R}$
, and rearranging the terms, we have
\begin{equation*} \sup_{t \in \mathbb{R}} \mathbb{E} [(\lambda_t )^p] \le \dfrac{(1+\epsilon)^{p-1} \nu^p + ((1+\epsilon) (1+\epsilon^{-1}) )^{p-1} K_p}{ 1 - (1+\epsilon^{-1})^{2p-2} \eta^p } < \infty.\end{equation*}
Remark A.1. Lemma A.1 can easily be extended to include all values of
$p \ge 2$
, but not for
$p \in (1,2)$
. To see this, by noting that the Burkholder–Davis–Gundy (BDG) inequality holds for
$p> 1$
, thus, when
$p \in (1,2)$
, the bound in (A.2) still holds by Lemma I.2.1.5 of [Reference Jacod and Protter23] (see also Lemma 10.2 of [Reference Clinet and Potiron6]). However, due to the quadratic variation term in the BDG inequality, we still require the integrability of
$g^2$
for the second term of (A.2) to be bounded in order to bound
$\sup_{t \in \mathbb{R}} \mathbb{E}[(\lambda_t)^p]$
. Hence the above proof is yet to cover the case when
$p \in (1,2)$
. On the other hand, for
$p \ge 2$
, no additional conditions are required to show that Lemma A.1 still holds for
$p \ge 2$
, and the proof follows readily from the arguments presented in the proof above. Furthermore, note that the bound in (A.2) requires an application of Lemma A.2 of [Reference Clinet and Yoshida7] with
$T = +\infty$
. The generalisation of Lemma A.2 to the case for
$T = +\infty$
holds as a consequence of the BDG inequality, which was used in its proof by the authors.
Lemma A.2. In addition to [C2], for any
$\theta \in \Theta$
, if
$ g(t;\,\theta)^p$
is integrable for some
$p \in \mathbb{N}$
, then
$\sup_{t \in \mathbb{R}} \sup_{\theta \in {\Theta}} \Vert \lambda_{t, \theta} \Vert_p<\infty$
and
$\sup_{t \in [0,n] } \sup_{\theta \in {\Theta}} \Vert \tilde{\lambda}_{t, \theta} \Vert_p<\infty$
Proof. We give details to prove the first case only since analogous statements can be made to show the second. Recall that
$\theta^0$
is the true parameter value such that
$\lambda_{t, \theta^0}$
denotes the true intensity of
$N_t$
and let
$\eta^0 = \eta(\theta^0)$
. Therefore, the proof to show that if
$g(t;\,\theta^0)^p$
is integrable, then
$\sup_{t \in \mathbb{R}} \Vert \lambda_{t, \theta^0} \Vert_p<\infty$
, is no different from the proof of Lemma A.1. Hence, it remains to generalise the result for any
$\theta$
, which we will do by induction on p again. For the initialising case when
$p = 1$
, note that
\begin{align*}\mathbb{E} [\lambda_{t, \theta} ] = \nu(\theta) + \int_{-\infty}^{t-} g(t-s;\, \theta) \mathbb{E}[ \lambda_{s, \theta^0} ] \,\mathrm{d} s \le \bar{\nu} + \bar{\eta} \sup_{t \in \mathbb{R}}\mathbb{E} [\lambda_{t, \theta^0} ]. \end{align*}
By taking the supremum in
$\theta$
over
$\Theta$
and rearranging the terms, it follows immediately that under [C2],
$\sup_{t \in \mathbb{R}} \sup_{\theta \in {\Theta}} \mathbb{E}[\lambda_{t, \theta}] < \infty$
. For
$p \ge 2$
, similarly to the bounds by (A.1) and (A.2),
$\mathbb{E}[ (\lambda_{t, \theta})^p ]$
is bounded by
\begin{align*} (1+\epsilon)^{p-1} \nu(\theta)^{p} + (1+\epsilon^{-1})^{p-1} \mathbb{E} \biggl[\biggl( \int_{-\infty}^{t-} g(t-s;\,\theta) \,\mathrm{d} N_s \biggr)^{p}\biggr], \end{align*}
and
\begin{align*}\mathbb{E} \biggl[\biggl( \int_{-\infty}^{t-} g(t-s;\,\theta) \,\mathrm{d} N_s \biggr)^{p}\biggr]\end{align*}
is bounded by
\begin{align} &(1+\epsilon)^{p-1} K_p \biggl\{ \mathbb{E} \biggl[\int_{-\infty}^0 g(t-s;\, \theta)^{p} \lambda_{s, \theta^0} \,\mathrm{d} s\biggr] + \mathbb{E} \biggl[\biggl( \int_{-\infty}^0 g(t-s;\, \theta)^{2} \lambda_{s, \theta^0} \,\mathrm{d} s \biggr)^{{p}/{2}} \biggr] \biggr\} \nonumber \\ &\quad + (1+\epsilon)^{p-1} K_p \biggl\{ \mathbb{E} \biggl[\int_0^{t-} g(t-s;\, \theta)^{p} \lambda_{s, \theta^0} \,\mathrm{d} s\biggr] + \mathbb{E} \biggl[\biggl( \int_0^{t-} g(t-s;\, \theta)^{2} \lambda_{s, \theta^0} \,\mathrm{d} s \biggr)^{{p}/{2}} \biggr] \biggr\} \nonumber \\ &\quad + (1+\epsilon^{-1})^{p-1} \mathbb{E} \biggl[ \biggl( \int_{-\infty}^{t-} g(t-s;\, \theta) \lambda_{s, \theta^0} \,\mathrm{d} s \biggr)^{p} \biggr]. \end{align}
Hence, similarly to the proof of Lemma A.1, by Hölder’s inequality, the induction hypothesis, and integrability of
$g(\cdot;\, \theta)^p$
, it can be shown that the first two lines of (A.4) are bounded by
$K_p$
uniformly in t and
$\theta$
, and
\begin{align*} \mathbb{E} \biggl[ \biggl( \int_{-\infty}^{t-} g(t-s;\, \theta) \lambda_{s, \theta^0} \,\mathrm{d} s \biggr)^{p} \biggr] \le \eta(\theta)^p \sup_{t \in \mathbb{R}}\mathbb{E}\bigl[ (\lambda_{t, \theta^0})^p\bigr]. \end{align*}
Collecting the above bounds, we have shown that
\begin{align*}& \mathbb{E}[ (\lambda_{t, \theta})^p] \\&\quad \le (1+\epsilon)^{p-1} \nu(\theta)^{p} + ( (1+\epsilon) (1+\epsilon^{-1}))^{p-1} K_p + ( 1 + \epsilon^{-1} )^{2p-2} \eta(\theta)^p \sup_{t \in \mathbb{R}} \mathbb{E} [ ( \lambda_{t, \theta^0})^p ]. \end{align*}
Taking successively the supremum in t and
$\theta$
, along with [C2](iii), we have the desired result.
Remark A.2 From the discussion in Remark A.1, it can also be shown that Lemma A.2 still holds for non-integer p for
$p \ge 2$
without any additional conditions.
Let
$a_n$
be a monotone sequence of positive real numbers that tends to infinity with n such that
$a_n = {\mathrm{o}} (n)$
. The following lemma is concerned with the difference between
$\lambda_{t, \theta}$
and
$\tilde{\lambda}_{t, \theta}$
for any
$\theta$
over
$[a_n, n]$
and is useful in proving Lemma 3.2.
Lemma A.3. In addition to [C2], for any
$\theta \in \Theta$
, if
$g(t;\, \theta)^p$
is integrable for some
$p \in \mathbb{N}$
, then
$\sup_{t \in [a_n, n]} \sup_{\theta \in \Theta} \Vert \lambda_{t, \theta} - \tilde{\lambda}_{t, \theta} \Vert_p = {\mathrm{o}} (1)$
.
Proof of Lemma
A.3. Similarly to the proof of Lemma A.2, let us consider the case when
$\theta = \theta^0$
first and prove by induction on p that
$\sup_{t \in [a_n, n]} \Vert \lambda_{t, \theta^0} - \tilde{\lambda}_{t, \theta^0} \Vert_p = {\mathrm{o}} (1)$
. When
$p = 1$
,
$\mathbb{E}[\vert \lambda_{t, \theta^0} - \tilde{\lambda}_{t, \theta^0} \vert ]$
is bounded by
\begin{equation*} \mathbb{E}\biggl[ \int_0^{t-} g(t-s;\, \theta^0) \vert \mathrm{d} N_s - \mathrm{d} \tilde{N}_s \vert \biggr] + \mathbb{E}\biggl[\int_{-\infty}^0 g(t-s;\, \theta^0) \,\mathrm{d} N_s \biggr]. \end{equation*}
By Poisson embedding and a fixed point argument (see e.g. [Reference Clinet and Potiron6]), N and
$\tilde{N}$
can be represented as an integral with respect to a unit rate Poisson process on
$\mathbb{R}^2$
such that they are both defined on the same probability space. For more details on Poisson embedding techniques, see [Reference Brémaud and Massoulié4], [Reference Brémaud and Massoulié5], or Section 14.7 of [Reference Daley and Vere-Jones14]. Let
$N^P$
denote a unit Poisson process on
$\mathbb{R}^2$
and let
$\mathcal{F}_t^{N^P}$
be the
$\sigma$
-field generated by the random variables
$N^P(C)$
for
$C \in \mathcal{B}([0, t] \times \mathbb{R})$
. Let
$\mathcal{F}^{N^P} = (\mathcal{F}_t^{N^P})_{t \in [0, n]}$
be a filtration induced by
$N^P$
on
$[0, n] \times \mathbb{R}$
and set
$\mathcal{F}_t = \mathcal{F}_t^{N^P}$
. By Poisson embedding, N and
$\tilde{N}$
have the following representations respectively:
\begin{align*} N_t & = \int_{(-\infty,t] \times \mathbb{R}} \unicode{x1D7D9}_{\{0 \le z \le \lambda_{s, \theta^0}\}} N^P (\mathrm{d} s \times \mathrm{d} z), \\ \tilde{N}_t & = \int_{[0, t] \times \mathbb{R}} \unicode{x1D7D9}_{\{0 \le z \le \tilde{\lambda}_{s, \theta^0}\}} N^P(\mathrm{d} s \times \mathrm{d} z). \end{align*}
By expressing N and
$\tilde{N}$
as integrals with respect to
$N^P$
,
$\vert \mathrm{d} N_s - \mathrm{d} \tilde{N}_s\vert$
is the integer measure that counts the jumps that do not belong to both
$ \mathrm{d} N_s$
and
$ \mathrm{d} \tilde{N}_s$
. This counting measure admits
$\vert \lambda_{s, \theta^0} - \tilde{\lambda}_{s, \theta^0} \vert$
as its stochastic intensity. Hence, by Fubini’s theorem,
$\mathbb{E} [\vert \lambda_{t, \theta^0} - \tilde{\lambda}_{t, \theta^0} \vert]$
is bounded by
\begin{equation*} \int_0^{t-} g(t-s;\, \theta^0) \mathbb{E}\bigl[\vert \lambda_{s, \theta^0} - \tilde{\lambda}_{s, \theta^0} \vert \bigr] \,\mathrm{d} s + \int_{-\infty}^0 g(t-s;\, \theta^0) \mathbb{E}[\lambda_{s, \theta^0} ] \,\mathrm{d}s. \end{equation*}
Recall that
$g = \eta \tilde{g}$
, where
$\tilde{g}$
is the density function of a non-negative random variable, and let
$\tilde{G}(t;\, \theta) = \int_0^t \tilde{g}(s;\, \theta)\,\mathrm{d} s$
denote the cumulative density function of
$\tilde{g}$
. By Lemma A.2, the second term above is bounded by
$K \eta^0 (1-\tilde{G}(t;\, \theta^0))$
, which converges to 0 as
$t \to \infty$
. Next, let
$(f \ast h )(t) = \int_0^t f(t-s) h(s) \,\mathrm{d} s$
denote the usual convolution, let
$h^{\ast (i)}$
be the ith convolution of h with itself, and let
$f(t) = \mathbb{E}[\vert \lambda_{t, \theta^0} - \tilde{\lambda}_{t, \theta^0} \vert ]$
; we can write
\begin{align*} f(t) & \le K \eta^0 ( 1- \tilde{G}(t;\,\theta^0) ) + (g(\cdot;\,\theta^0) \ast f(\cdot) ) (t) \\ & \le K \eta^0 (1 - \tilde{G}(t;\,\theta^0) ) + K \eta^0 \sum_{i=1}^j ( (1 - \tilde{G}(\cdot;\,\theta^0) ) \ast g^{\ast (i)} (\cdot;\,\theta^0) ) (t) \\ & \quad + ( g^{\ast (j)}(\cdot;\,\theta^0) \ast f(\cdot) )(t). \end{align*}
Here
$\tilde{G}^{\ast (i)}$
denotes the distribution function of the ith convolution of some non-negative random variable with distribution function
$\tilde{G}$
. Hence, by noting that
$\tilde{G}^{\ast (j)}(\cdot; \,\theta) \le \tilde{G}^{\ast (i)}(\cdot; \, \theta)$
for all
$j \ge i$
, we have
\begin{align*} \sum_{i=1}^j \bigl( \bigl(1 - \tilde{G}(\cdot;\,\theta^0) \bigr) \ast g^{\ast (i)} (\cdot;\,\theta^0) \bigr) (t) & = \sum_{i=1}^j \bigl(\eta^0 \bigr)^i \bigl(\tilde{G}^{\ast (i)} (t;\,\theta^0) - \tilde{G}^{\ast (i+1)}(t;\,\theta^0)\bigr) \\ & \le \bigl(1- \tilde{G}^{\ast (j+1)} (t;\,\theta^0) \bigr) \sum_{i=1}^j \bigl(\eta^0 \bigr)^i. \end{align*}
By Lemma A.2 again, we have
\begin{align*} \bigl( g^{\ast (j)}(\cdot;\,\theta^0) \ast f(\cdot) \bigr) (t) & = \int_0^t \bigl(\eta^0\bigr)^j \tilde{g}^{\ast (j)}(t-s;\,\theta^0) \mathbb{E} \bigl[ \vert \lambda_{s, \theta^0} - \tilde{\lambda}_{s, \theta^0} \vert \bigr] \,\mathrm{d} s \\ & \le K \bigl(\eta^0\bigr)^j \tilde{G}^{\ast (j)}(t;\,\theta^0) \\ & \le K \bigl(\eta^0 \bigr)^j. \end{align*}
By collecting the derived bounds above, we deduce that for any
$\epsilon >0$
there exists a j sufficiently large that
$K(\eta^0)^j \le \epsilon$
and
\begin{align*} f(t) & \le K \eta^0 \bigl(1 - \tilde{G}(t;\,\theta^0) \bigr) + K \eta^0 \bigl(1 - \tilde{G}^{\ast (j)} (t;\,\theta^0) \bigr) \sum_{i=1}^j (\eta^0)^i + \epsilon. \end{align*}
From here, it is straightforward to see that for some sufficiently large n, f(t) can be made arbitrarily small after taking the supremum in t over
$[a_n, n]$
.
For
$p >1$
, let
$f(t) = \mathbb{E}[\vert \lambda_{t, \theta^0} - \tilde{\lambda}_{t, \theta^0}\vert^p]$
. Then, similarly to the proofs of Lemmas A.1 and A.2, we can bound f(t) by
\begin{equation*} (1+\epsilon^{-1})^{p-1} \mathbb{E} \biggl[ \biggl\vert \int_0^{t-} g(t-s;\, \theta^0) (\mathrm{d} N_s - \mathrm{d} \tilde{N}_s ) \biggr\vert^p \biggr] + \bigl(1+\epsilon\bigr)^{p-1} \mathbb{E}\biggl[\biggl( \int_{-\infty}^0 g(t-s;\, \theta^0) \,\mathrm{d} N_s \biggr)^p\biggr], \end{equation*}
where
\begin{align*}\mathbb{E}\biggl[\biggl\vert \int_0^{t-} g(t-s;\, \theta^0) (\mathrm{d} N_s - \mathrm{d} \tilde{N}_s)\biggr\vert^p\biggr]\end{align*}
is bounded by
\begin{align*} K_p \bigg\{ &\mathbb{E}\biggl[ \int_0^{t-} g(t-s;\, \theta^0)^p \vert \lambda_{s, \theta^0} - \tilde{\lambda}_{s, \theta^0} \vert \,\mathrm{d} s\biggr] + \mathbb{E}\biggl[\biggl( \int_0^{t-} g(t-s;\, \theta^0)^2 \vert \lambda_{s, \theta^0} - \tilde{\lambda}_{s,\theta^0} \vert \,\mathrm{d} s \biggr)^{{p}/{2}}\biggr]\\ & + \mathbb{E}\biggl[ \biggl(\int_0^{t-} g(t-s;\, \theta^0) \vert \lambda_{s, \theta^0} - \tilde{\lambda}_{s,\theta^0} \vert \,\mathrm{d} s \biggr)^p \biggr] \bigg\}. \end{align*}
By the induction hypothesis, for any
$\delta >0$
and
$j \in \{2, \ldots, p-1\}$
, there exists
$\tilde{t} >0$
sufficiently large that
$\mathbb{E} [ \vert \lambda_{t, \theta^0} - \tilde{\lambda}_{t, \theta^0} \vert^j] < \delta$
whenever
$t > \tilde{t}$
. Thus, by Lemma A.2, an application of Hölder’s inequality similar to the one used in the proof of Lemma A.1, and Fubini’s theorem, for
$i \ge 2$
and
$2 \le j < p$
, we have
\begin{align*} &\mathbb{E} \biggl[ \biggl( \int_0^{t-} g(t-s;\, \theta^0)^i \vert \lambda_{s, \theta^0} - \tilde{\lambda}_{s, \theta^0} \vert \,\mathrm{d} s \biggr)^j \biggr]\\ & \quad \le \biggl(\int_0^{t-} g(t-s;\, \theta^0) ^i \,\mathrm{d} s \biggr)^{j-1} \int_0^{t-} g(t-s;\, \theta^0)^i \mathbb{E}\bigl[\vert \lambda_{s, \theta^0} - \tilde{\lambda}_{s, \theta^0} \vert^j \bigr] \,\mathrm{d} s\\ & \quad = \biggl(\int_0^{t-} g(t-s;\, \theta^0) ^i \,\mathrm{d} s \biggr)^{j-1} \bigg\{ \int_0^{\tilde{t}} g(t-s;\, \theta^0)^i \mathbb{E}\bigl[\vert \lambda_{s, \theta^0} - \tilde{\lambda}_{s, \theta^0} \vert^j \bigr] \,\mathrm{d} s \\ & \quad \quad + \int_{\tilde{t}}^{t-} g(t-s;\, \theta^0)^i \mathbb{E}\bigl[\vert \lambda_{s, \theta^0} - \tilde{\lambda}_{s, \theta^0} \vert^j \bigr] \,\mathrm{d} s \bigg\} \\ & \quad \le K_j \int_0^{\tilde{t}} g(t-s;\, \theta^0)^i \,\mathrm{d} s + \delta K \int_{\tilde{t}}^{t-} g(t-s;\, \theta^0)^i \,\mathrm{d} s. \end{align*}
Recall that if
$g^p$
is integrable, then
$g^q$
is also integrable for
$q \in [1, p]$
. From this, we deduce that
\begin{align*}\int_0^{\tilde{t}} g(t-s;\,\theta^0)^i \,\mathrm{d} s \to 0\quad \text{as $t - \tilde{t} \to \infty$},\end{align*}
and
\begin{align*}\int_{\tilde{t}}^{t-} g(t-s;\, \theta^0)^i \,\mathrm{d} s < \infty\quad \text{for $i \le p$.}\end{align*}
This shows that
\begin{align*}\mathbb{E}\biggl[\biggl\vert \int_0^{t-} g(t-s;\, \theta^0) (\mathrm{d} N_s - \mathrm{d} \tilde{N}_s)\biggr\vert^p\biggr] \to 0\quad \text{as $t \to \infty$.}\end{align*}
Next, we show that
\begin{align*}\mathbb{E}\biggl[\biggl( \int_{-\infty}^0 g(t-s;\, \theta^0) \,\mathrm{d} N_s\biggr)^p\biggr] \to 0\quad \text{as $t \to \infty$.}\end{align*}
Note that
\begin{align*}\mathbb{E}\biggl[\biggl( \int_{-\infty}^0 g(t-s;\, \theta^0) \,\mathrm{d} N_s\biggr)^p\biggr]\end{align*}
is bounded by
\begin{align*}& K_p \biggl\{ \mathbb{E}\biggl[ \int_{-\infty}^0 g(t-s;\, \theta^0)^p \lambda_{s, \theta^0} \,\mathrm{d} s \biggr] + \mathbb{E}\biggl[ \biggl(\int_{-\infty}^0 g(t-s;\, \theta^0)^2 \lambda_{s, \theta^0} \,\mathrm{d} s \biggr)^{{p}/{2}}\biggr] \nonumber \\ & \quad + \mathbb{E}\biggl[ \biggl(\int_{-\infty}^0 g(t-s;\, \theta^0)^p \lambda_{s, \theta^0} \,\mathrm{d} s \biggr)^p \biggr]\biggr\}. \end{align*}
By Lemma A.2, Hölder’s inequality, and Fubini’s theorem, for
$i \ge 2$
and
$j \ge 2$
, we have
\begin{align*} & \mathbb{E}\biggl[\biggl(\int_{-\infty}^0 g(t-s;\, \theta^0)^i \lambda_{s, \theta^0} \,\mathrm{d} s \biggr)^j \biggr] \\ & \quad \le \biggl(\int_{-\infty}^0 g(t-s;\, \theta^0)^i \,\mathrm{d} s \biggr)^{j-1} \int_{-\infty}^0 g(t-s;\, \theta^0)^i \mathbb{E}\bigl[ \bigl(\lambda_{s, \theta^0} \bigr)^j \bigr] \,\mathrm{d} s\\ & \quad \le K_j \biggl(\int_{-\infty}^0 g(t-s;\,\theta^0)^i \,\mathrm{d} s \biggr)^j. \end{align*}
Furthermore, we can deduce from the integrability of
$g(\cdot;\,\theta^0)^p$
that
\begin{align*}\int_{-\infty}^0 g(t-s;\, \theta^0)^i\,\mathrm{d} s \to 0\quad \text{as $t \to \infty$}\end{align*}
for any
$i \le p$
. In view of this and the above bound, we conclude that
\begin{align*}\mathbb{E}\biggl[ \biggl(\int_{-\infty}^0 g(t-s;\, \theta^0)^i \lambda_{s, \theta^0} \,\mathrm{d} s \biggr)^j\biggr] \to 0\quad \text{as $t \to \infty$.}\end{align*}
This shows that under [C2], if
$g(t;\,\theta)^p$
is integrable for all
$\theta \in \Theta$
, then
\begin{align*} \sup_{t \in [a_n, n]} \Vert \lambda_{t, \theta^0} - \tilde{\lambda}_{t, \theta^0} \Vert_p = {\mathrm{o}} (1). \end{align*}
To this end, we have only considered the case when
$\theta = \theta^0$
. The proof to show that the statement holds for any
$\theta \in \Theta$
is analogous to the proof of Lemma A.2, and we omit the details here.
A.2. Proofs of Section 2
Proof of Lemma
2.1. By Lemma 1 of [Reference Hawkes and Oakes21], under [C1], there exists a stationary Poisson cluster process representation of N where the process of cluster centres is the stationary Poisson process with rate
$\nu$
. Each cluster centre is associated with a component process and the component processes are mutually independent by construction. By Proposition 12.3.IX of [Reference Daley and Vere-Jones14], a stationary cluster process is mixing, in the sense of Definition 2.2, whenever the cluster centre process has the same property. Since the cluster centre process of N is the stationary Poisson process and is therefore mixing, we deduce that N is also mixing.
Proof of Lemma 2.2. Since
$\lambda_t$
is stationary, it suffices to show that for any
$\phi$
and
$\psi \in C_b(\mathbb{R}_+, \mathbb{R})$
,
\begin{equation*} \mathrm{cov} (\phi(\lambda_t), \psi(\lambda_{t +u}) ) \rightarrow 0, \quad \vert u \vert \rightarrow \infty. \end{equation*}
Equivalently, it is sufficient to show instead that
\begin{equation} \lim_{u \rightarrow \infty}\mathbb{E} [\phi(\lambda_{t}) \psi(\lambda_{t+u})] = \mathbb{E} [\phi(\lambda_{t} )] \mathbb{E} [\psi(\lambda_{t})]. \end{equation}
Note that we can express the intensity of
$N_t$
as
\begin{align*} \lambda_{t} & = \nu + \int_{-\infty}^{\infty} \unicode{x1D7D9}_{(-\infty, t)}(s) g(t-s)\,\mathrm{d} N_s. \end{align*}
Similarly, the intensity of
$N_{t+u}$
can be written as
\begin{align*} \lambda_{t+u} & = \nu + \int_{-\infty}^{\infty} \unicode{x1D7D9}_{(-\infty, t)}(s) g(t-s)\,\mathrm{d} ( S_u N_s ). \end{align*}
A counting measure is boundedly finite if only a finite number of points fall inside any bounded set. Let
$\mathcal{N}$
denote the space of all boundedly finite counting measures defined on the Borel sets
$\mathcal{B}(\mathbb{R})$
of
$\mathbb{R}$
and
$\mathcal{B}(\mathcal{N})$
the corresponding
$\sigma$
-algebra. Let
$f\colon \mathcal{B}( \mathcal{N}) \to \mathbb{R}_+$
with
\begin{align*}f(V) = \nu + \int_{-\infty}^\infty \unicode{x1D7D9}_{(-\infty, t)}(s) g(t-s) \,\mathrm{d} V_s.\end{align*}
Note that for any
$\phi$
and
$\psi \in C_b(\mathbb{R}_+, \mathbb{R})$
, there exist bounded and measurable functions
$\tilde{\phi} = \phi \circ f$
and
$\tilde{\psi} = \psi \circ f$
that map
$\mathcal{B}(\mathcal{N})$
to
$\mathbb{R}$
such that
$\phi(\lambda_t) \psi(\lambda_{t+u}) = \tilde{\phi}(N) \tilde{\psi}(S_uN)$
. Let
$\tilde{\phi}^+$
,
$\tilde{\phi}^-$
,
$\tilde{\psi}^+$
, and
$\tilde{\psi}^-$
be non-negative, bounded, and measurable functions on
$\mathcal{B} (\mathcal{N})$
such that
$\tilde{\phi} = \tilde{\phi}^+ - \tilde{\phi}^-$
, and
$\tilde{\psi} = \tilde{\psi}^+ - \tilde{\psi}^-$
. By the simple function approximation theorem (see e.g. [Reference Royden and Fitzpatrick36, page 62] or [Reference Kallenberg24, Lemma 1.12]), there exist increasing sequences of simple functions
$\{\tilde{\phi}^+_i\}_{i=1}^\infty$
,
$\{\tilde{\phi}_i^-\}_{i=1}^\infty$
,
$\{\tilde{\psi}_i^+\}_{i=1}^\infty$
and
$\{\tilde{\psi}_i^-\}_{i=1}^\infty$
such that
$\tilde{\phi}^+_i$
,
$\tilde{\phi}^-_i$
,
$\tilde{\psi}^+_i$
, and
$\tilde{\psi}^-_i$
all converge pointwise to
$\tilde{\phi}^+$
,
$\tilde{\phi}^-$
,
$\tilde{\psi}^+$
, and
$\tilde{\psi}^-$
respectively. Furthermore, since
$\tilde{\phi}^+$
,
$\tilde{\phi}^-$
,
$\tilde{\psi}^+$
, and
$\tilde{\psi}^-$
are all bounded, the convergence is uniform in
$\mathcal{B} (\mathcal{N})$
. By Lemma 1.13 of [Reference Kallenberg24],
$(\tilde{\phi}^+_i - \tilde{\phi}^-_i ) ( \tilde{\psi}^+_i - \tilde{\psi}^-_i)$
converges uniformly to
$(\tilde{\phi}^+ - \tilde{\phi}^-)(\tilde{\psi}^+ - \tilde{\psi}^-)$
on
$\mathcal{B} (\mathcal{N})$
. This permits the application of the dominated convergence theorem such that
\begin{align} &\lim_{u \rightarrow \infty} \mathbb{E}\Bigl[{\lim_{i \rightarrow\infty} \bigl( \tilde{\phi}^+_i(N) - \tilde{\phi}^-_i (N) \bigr) \bigl(\tilde{\psi}^+_i(S_u N) - \tilde{\psi}^-_i(S_u N) \bigr)}\Bigr] \nonumber\\ &\quad =\lim_{u \rightarrow \infty} \lim_{i \rightarrow\infty}\mathbb{E}\bigl[{\bigl( \tilde{\phi}^+_i(N) - \tilde{\phi}^-_i (N) \bigr) \bigl(\tilde{\psi}^+_i(S_u N) - \tilde{\psi}^-_i(S_u N) \bigr)}\bigr]. \end{align}
Since
$(\tilde{\phi}^+_i - \tilde{\phi}^-_i ) ( \tilde{\psi}^+_i - \tilde{\psi}^-_i)$
converges uniformly to
$(\tilde{\phi}^+ - \tilde{\phi}^-)(\tilde{\psi}^+ - \tilde{\psi}^-)$
and the expectation in (A.6) is taken with respect to simple functions, we can interchange the order of the limit operators [Reference Rudin37, Theorem 7.11] in (A.6). In view of the mixing property of N from Lemma 2.1 and by the dominated convergence theorem, (A.6) can be written as
\begin{align*} & \lim_{i \rightarrow\infty} \lim_{u \rightarrow \infty} \mathbb{E}\bigl[{\bigl( \tilde{\phi}^+_i(N) - \tilde{\phi}^-_i (N) \bigr) \bigl(\tilde{\psi}^+_i(S_u N) - \tilde{\psi}^-_i(S_u N) \bigr)}\bigr]\\ & \quad = \lim_{i \rightarrow\infty} \mathbb{E}\bigl[{ \tilde{\phi}^+_i(N) - \tilde{\phi}^-_i (N) }\bigr] \mathbb{E}\bigl[{\tilde{\psi}^+_i(N) - \tilde{\psi}^-_i(N) }\bigr]\\ & \quad = \mathbb{E}[{\tilde{\phi}(N)}]\mathbb{E}[{\tilde{\psi}(N)}] \\ & \quad = \mathbb{E}[{\phi(\lambda_{t})}]\mathbb{E}[{\psi(\lambda_{t})}], \end{align*}
which is equivalent to (A.5).
Proof of Lemma 2.3. Since ergodicity requires convergence in probability, it is sufficient to consider the mean square error of
\begin{align*}\dfrac{1}{T}\int_0^T\psi(\lambda_t)\,\mathrm{d} t\end{align*}
and show that it converges to 0 as
$T \to \infty$
. By the definition of
$\pi$
and stationarity of
$\lambda$
, we have
\begin{align*} \mathbb{E} \biggl[ \dfrac{1}{T}\int_0^T \psi(\lambda_t) \,\mathrm{d} t - \pi(\psi)\biggr] = 0. \end{align*}
Hence, by Fubini’s theorem, we have
\begin{align} \quad + \dfrac{1}{T^2} \sup_{ \vert t - s\vert \le \sqrt{T}} \mathrm{cov} (\psi(\lambda_t), \psi(\lambda_s) ) \int_0^T \int_0^T \unicode{x1D7D9}_{\{\vert t - s \vert \le \sqrt{T} \} } \,\mathrm{d} s \,\mathrm{d} t. \end{align}
By Lemma 2.2, (A.7) converges to 0 as
$T \to \infty$
, and (A.8) is of order
${\mathrm{O}} (1/{\sqrt{T}})$
.
A.3. Proofs of Section 3
Proof of Lemma
3.1. By Lenglart’s inequality [Reference Andersen, Borgan, Gill and Keiding2, II.5.2.1, page 86], for any
$\epsilon >0$
and
$\delta >0$
, we can write
\begin{equation} \mathbb{P}\biggl( \sup_{t \in [0, n]} \bigg\vert \dfrac{1}{n} \int_0^t \log \biggl( \dfrac{\tilde{\lambda}_{s, \theta}}{\tilde{\lambda}_{s, \theta^0}} \biggr) \,\mathrm{d} \bar{\tilde{N}}_s \bigg\vert > \epsilon \biggr) \le \dfrac{\delta}{\epsilon^2} + \mathbb{P} \biggl( \dfrac{1}{n^2} \int_0^n \log \biggl( \dfrac{\tilde{\lambda}_{s, \theta}}{\tilde{\lambda}_{s, \theta^0}} \biggr)^2 \tilde{\lambda}_{s, \theta^0} \,\mathrm{d} s > \delta \biggr). \end{equation}
By Lemma A.2 and Hölder’s inequality, we have
\begin{equation*} \bigg\Vert \dfrac{1}{n^2} \int_0^n \log \biggl( \dfrac{\tilde{\lambda}_{s, \theta}}{\tilde{\lambda}_{s, \theta^0}} \biggr)^2 \tilde{\lambda}_{s, \theta^0} \,\mathrm{d} s \bigg\Vert \le \dfrac{1}{n} \sup_{s \in [0,n]} \bigg\Vert \log \biggl( \dfrac{\tilde{\lambda}_{s, \theta}}{\tilde{\lambda}_{s, \theta^0}} \biggr)^2 \bigg\Vert_2 \Vert \tilde{\lambda}_{s, \theta^0} \Vert_2 = {\mathrm{o}} (1). \end{equation*}
Hence, for any
$\delta > 0$
, the second term of the bound in (A.9) converges to 0 as n tends to infinity. Furthermore, the first term of the bound in (A.9) can be made arbitrarily small by controlling
$\epsilon$
and
$\delta$
. This shows that
$\tilde{M}^n(\theta, \theta^0) \to 0$
in probability.
Proof of Lemma
3.2. For some monotone sequence of positive reals
$a_n$
such that
$a_n = {\mathrm{o}} (n)$
, by Lemma A.2 we can replace the lower limit of the integral in
$\vert \tilde{L}^n(\theta, \theta^0) - L^n(\theta, \theta^0)\vert$
with
$a_n$
such that it is asymptotically equivalent to
\begin{align*} & \biggl\vert\dfrac{1}{n} \int_{a_n}^n \log(\tilde{\lambda}_{t, \theta} ) \tilde{\lambda}_{t, \theta^0} - \tilde{\lambda}_{t, \theta} - \log(\lambda_{t, \theta} ) \lambda_{t, \theta^0} + \lambda_{t, \theta} \,\mathrm{d} t \bigg\vert \\ &\quad \le \dfrac{1}{n} \int_{a_n}^n \vert \log ( {\tilde{\lambda}_{t, \theta}}/{\lambda_{t, \theta}}) \tilde{\lambda}_{t, \theta^0} \vert + \vert \log(\lambda_{t, \theta} ) (\tilde{\lambda}_{t,\theta^0} - \lambda_{t, \theta^0} ) \vert + \vert \tilde{\lambda}_{t, \theta} - \lambda_{t, \theta} \vert \,\mathrm{d} t. \end{align*}
By Lemmas A.2 and A.3, Hölder’s inequality, and the inequality that
\begin{align*}\vert \log(x+1) \vert \le \biggl\vert \dfrac{x}{\sqrt{x + 1}} \biggr\vert\quad\text{for $x > -1$,}\end{align*}
we deduce if
$g(t;\, \theta)^2$
is integrable, then
\begin{align*} \sup_{t \in [a_n, n]} \bigl\Vert \log\bigl( \tilde{\lambda}_{t, \theta} / \lambda_{t, \theta} \bigr) \tilde{\lambda}_{t, \theta^0} \bigr\Vert & \le \sup_{t \in [a_n, n]} \bigg\Vert \bigl(\tilde{\lambda}_{t, \theta} - \lambda_{t, \theta} \bigr) \dfrac{\tilde{\lambda}_{t, \theta^0}}{\sqrt{\tilde{\lambda}_{t, \theta} \lambda_{t, \theta^0}}} \bigg\Vert \\ & \le \dfrac{1}{\underline{\nu}} \sup_{t \in [a_n, n]} \Vert \tilde{\lambda}_{t, \theta} - \lambda_{t, \theta} \Vert_2 \Vert \tilde{\lambda}_{t, \theta^0} \Vert_2\\ &= \mathrm{o} (1),\\ \sup_{t \in [a_n, n]} \bigl\Vert \log( \lambda_{t, \theta} ) \bigl(\tilde{\lambda}_{t, \theta^0} - \lambda_{t, \theta^0} \bigr) \bigr\Vert & \le \sup_{t \in [a_n, n]} \Vert \log(\lambda_{t, \theta}) \Vert_2 \Vert \tilde{\lambda}_{t, \theta} - \lambda_{t, \theta} \Vert _2 \\ & = \mathrm{o} (1), \end{align*}
and
$\sup_{t \in [a_n, n]}\Vert \tilde{\lambda}_{t, \theta} - \lambda_{t, \theta}\Vert = {\mathrm{o}} (1)$
. From here, it is straightforward to see that for any
$\theta \in \Theta$
,
$\Vert \tilde{L}^n(\theta, \theta^0) - L^n(\theta, \theta^0) \Vert \to 0$
.
Proof of Lemma
3.3. Since
$\pi(\theta, \theta^0)$
is continuous in
$\theta$
and
$\Theta$
is compact, we can prove the result by contradiction by supposing that there exists
$\epsilon >0$
such that
$\pi(\theta, \theta^0) = 0$
for some
$\theta \in \Theta \backslash B(\theta^0, \epsilon)$
. Then, in view of (3.4), we have
\begin{equation*} \mathbb{E} [ \log(\lambda_{t, \theta} / \lambda_{t, \theta^0} ) \lambda_{t, \theta^0} - \lambda_{t, \theta} + \lambda_{t, \theta^0}] = 0. \end{equation*}
Since
$\log(x/y)y - x + y = 0$
if and only if
$x = y$
, we deduce
$\log(\lambda_{t, \theta} / \lambda_{t, \theta^0} ) \lambda_{t, \theta^0} - \lambda_{t, \theta} + \lambda_{t, \theta^0} = 0$
almost surely. This implies that for any
$t \in \mathbb{R}$
,
$\lambda_{t, \theta} = \lambda_{t, \theta^0}$
almost surely. By [C3], we have a contradiction.
Acknowledgements
We would like to thank the Editor-in-Chief Professor Peter Taylor and the anonymous referee for their helpful comments and advice which improved the content and presentation of this work.
Funding information
Kwan was supported by the Australian government research training program scholarship and the faculty top-up scholarships from the School of Mathematics and Statistics, UNSW Sydney. Chen was partly supported by a UNSW SFRG grant PS35307.
Competing interests
There were no competing interests to declare which arose during the preparation or publication process of this article.
