1. Introduction and main results
Let
$X=(X_{t},t\geq0)$
be a continuous-time birth–death process taking values in
$\mathbb{Z}_{+}\,:\!=\{0\}\cup\mathbb{N}$
, where 0 is an absorbing state and
$\mathbb{N}=\{1,2,\ldots\}$
is an irreducible transient class. Its jump rate matrix
$(q_{ij},i,j\in\mathbb{Z}_{+})$
satisfies
\begin{equation*} q_{ij}=\left\{\begin{array}{l@{\quad}l} b_i&{\textrm{if}}\ \ j=i+1,i\geq0,\\[4pt] d_i&{\textrm{if}}\ \ j=i-1,i\geq1,\\[4pt] -(b_i+d_i)&{\textrm{if}}\ \ j=i,i\geq0,\\[4pt] 0&{\textrm{otherwise}}, \end{array}\right.\end{equation*}
where the birth rates
$(b_i$
,
$i\in\mathbb{N})$
and death rates
$(d_i$
,
$i\in\mathbb{N})$
are strictly positive, and
$d_{0}=b_{0}=0$
.
Consider
$\pi=(\pi_i,i\in\mathbb{N})$
with the coefficients
\begin{equation} \pi_1=1 , \qquad \pi_i=\frac{{{b}}_{1}{{b}}_{2}\cdots{{b}}_{i-1}}{{{d}}_{2}{{d}}_{3}\cdots{{d}}_{i}}, \quad i\geq2.\end{equation}
Then, we have
$b_i\pi_i=d_{i+1}\pi_{i+1}$
for
$i\in\mathbb{N}$
, which implies that the process X is reversible with respect to
$\pi$
, that is, for all
$i,j \in \mathbb{N}$
,
$\pi_iq_{ij}=\pi_jq_{ji}$
. Put
\begin{equation} A=\sum_{i=1}^\infty\frac{1}{b_i\pi_i},\quad B=\sum_{i=1}^\infty{\pi}_{i}, \quad R=\sum_{i=1}^\infty\frac{1}{b_i\pi_i}\sum_{j=1}^{i}\pi_{j}, \quad S=\sum_{i=1}^\infty\frac{1}{b_i\pi_i}\sum_{j=i+1}^{\infty}\pi_{j} , \end{equation}
so we have
\begin{equation*} R+S=AB, \qquad A=\infty\Rightarrow R=\infty, \qquad S<\infty\Rightarrow B<\infty.\end{equation*}
In this paper, we assume that the birth–death process is surely killed at 0, that is, for all
$i\in \mathbb{N}$
,
$\mathbb{P}_{i}(T_{0}<\infty)=1$
, where
$T_{0}=\inf\{t\geq0 \,:\, X_t=0\}$
is the absorption time of the process X, and
$\mathbb{P}_{i}$
denotes the probability measure of the process when the initial state is i. It is well known (see, e.g., [Reference Karlin and McGregor10]) that the assumption on sure killing,
$\mathbb{P}_{i}(T_{0}<\infty)=1$
for all
$i\in\mathbb{N}$
, is equivalent to
$A=\infty$
. Note that
$A=\infty$
implies the process X is non-explosive. Let
$(P_{t})_{t\geq0}$
be the semigroup of the process X before killing at 0. Then, for all
$i\in \mathbb{N}$
,
$P_{t}\,f(i)=\mathbb{E}_{i}[f(X_{t}),T_{0}>t]$
, and it acts on the set of bounded measurable functions defined on
$\mathbb{N}$
. Here,
$\mathbb{E}_{i}$
denotes the expectation with respect to
$\mathbb{P}_{i}$
.
For such a process, we are interested in the asymptotic behavior of the process conditioned on long-term survival. A well-studied object (see, e.g., [Reference Collet, Martnez and San Martn6, Reference Méléard and Villemonais12]) is the quasi-stationary distribution, that is, a probability measure
$\alpha$
on
$\mathbb{N}$
such that, for any
$t\geq0$
,
\begin{equation}\mathbb{P}_{\alpha}(X_{t}\in\cdot|T_{0}>t)=\alpha.\end{equation}
Here, as usual,
$\mathbb{P}_{\alpha}\,:\!=\,\sum_{i\in\mathbb{N}}\alpha_{i}\mathbb{P}_{i}$
. If there exists a probability measure
$\mu$
on
$\mathbb{N}$
such that
\begin{equation}\lim\limits_{t\rightarrow\infty}\mathbb{P}_{\mu}(X_{t}\in\cdot|T_{0}>t)=\alpha,\end{equation}
then we say that
$\mu$
is attracted to
$\alpha$
, or is in the domain of attraction of
$\alpha$
, for the conditional evolution. For any bounded and measurable function f on
$\mathbb{N}$
, (1.3) can also be written as
\begin{equation*}\frac{\alpha(P_{t}f)}{\alpha(P_{t}\textbf{1})}=\alpha(f),\end{equation*}
where
$\textbf{1}=\unicode{x1d7d9}_{\mathbb{N}}$
and
$\alpha(\,f)=\sum_{i\in\mathbb{N}}\alpha_{i}\,f(i)$
.
The existence, uniqueness, and other properties of quasi-stationary distributions for birth–death processes has been extensively studied in past decades. On quasi-stationary distributions of birth–death processes, van Doorn [Reference van Doorn16] gave the following picture of the situation: there is no quasi-stationary distribution, a unique quasi-stationary distribution, or an infinite continuum of quasi-stationary distributions. Zhang and Zhu [Reference Zhang and Zhu18] proved that the unique quasi-stationary distribution attracts all initial distributions supported in
$\mathbb{N}$
. Villemonais [Reference Villemonais17] provided some new results on the domain of attraction of the minimal quasi-stationary distribution. Until now, no one has completely solved the problem of the domains of attraction of the infinite continuum of quasi-stationary distributions for birth–death processes. However, existence and uniqueness of a quasi-stationary distribution or attraction of all initial distributions do not imply uniform convergence. This paper is devoted to studying the speed of convergence of the conditional probability measure
$\mathbb{P}_{\mu}(X_{t}\in\cdot|T_{0}>t)$
, for some initial measures
$\mu$
on
$\mathbb{N}$
, towards the quasi-stationary distribution
$\alpha$
when t goes to infinity. The total variation distance is usually used to quantify the weak convergence (1.4) – see, e.g., [Reference Champagnat and Villemonais2, Reference Champagnat and Villemonais3, Reference Martnez, San Martn and Villemonais11] – defined as
$\|\mu-\nu\|_{\textrm{TV}}\,:\!=\,\sup_{f\in\mathcal{B}_{1}(\mathbb{N})}|\mu(f)-\nu(f)|$
, where
$\mu, \nu$
are any two probability measures on
$\mathbb{N}$
,
$\mathcal {B}_{1}(\mathbb{N})$
denotes the set of bounded measurable functions defined on
$\mathbb{N}$
such that
$\|\,f\|_{\infty}\leq1$
, and
$\|f\|_{\infty}=\sup_{i\in\mathbb{N}}|f(i)|$
. Other distances, for example the 1-Wasserstein distance [Reference Oçafrain13], can also be used to quantify the weak convergence (1.4).
The boundary point
$\infty$
is called the entrance boundary if
$R=\infty$
,
$S<\infty$
. When
$\infty$
is an entrance boundary, the following result was obtained by Martnez et al. [Reference Martnez, San Martn and Villemonais11, Theorem 2]; this work proves that implication
$(\textrm{ii})$
implies
$(\textrm{iii})$
by using the h-transform (or Doob’s h-transform). Here, h is the eigenfunction of the first nontrivial eigenvalue of the infinitesimal operator of the original absorbed process.
Theorem 1.1. For the birth–death process X satisfying
$A=\infty$
, the following statements are equivalent:
-
(i)
$S<\infty$
. -
(ii) There exists a unique quasi-stationary distribution.
-
(iii) There exist a probability measure
$\alpha$
and two constants
$C^{\prime}, \gamma>0$
such that, for all
$t\geq0$
and all probability measures
$\mu$
on
$\mathbb{N}$
,
$\left\|\mathbb{P}_{\mu}(X_t\in \cdot|T_{0}>t)-\alpha\right\|_{\textrm{TV}}\leq C^{\prime}\textrm{e}^{-\gamma t}$
.
Moreover, in
$(\textrm{iii})$
the distribution
$\alpha$
is the unique quasi-stationary distribution of X.
It is already known (see [Reference van Doorn16]) that (i) and (ii) are equivalent. This equivalence is only mentioned here for completeness. For any measurable function
$\psi\,:\,\mathbb{N}\rightarrow[1,+\infty)$
, the
$\psi$
-norm of a signed measure
$\mu$
is defined as
$\|\mu\|_{\psi}\,:\!=\,\sup_{|f|\leq\psi}|\mu(f)|$
. If
$\psi=1$
, then the
$\psi$
-norm is the total variation norm. Let
$\|{\cdot}\|_{2}$
be the
$\mathbb{L}^2(m)$
-norm, defined by
$\|f\|_{2}=\big(\!\sum_{i\in\mathbb{N}}m_{i}f^{2}(i)\big)^{{1}/{2}}$
. Here, m is the unique stationary distribution of the Q-process Y defined in Section 2. For convenience, we denote
\begin{equation} \eta(i)\,:\!=\,Q_{i}(\lambda_{c}) , \qquad \eta\circ\mu_{i}\,:\!=\,\frac{\mu_{i}\eta(i)}{\mu(\eta)},\end{equation}
where
$Q_{i}(\lambda_{c})$
is defined in Section 2 and denotes the eigenfunction of the first nontrivial eigenvalue of the infinitesimal operator of the process X. We write
$\mathbb{E}_{\mu}$
for the expectation with respect to
$\mathbb{P}_{\mu}$
. Further, we have the following result.
Theorem 1.2. Let X be a birth and death process satisfying
$A=\infty$
and
$S<\infty$
. Assume that there exists a function
$\psi\,:\,\mathbb{N}\rightarrow[1,+\infty)$
such that
$\alpha(\psi^{2})<+\infty$
, where
$\alpha$
is the unique quasi-stationary distribution of X. Then, for any probability measure
$\mu$
on
$\mathbb{N}$
, there exist
$t_{\mu}$
and
$\varepsilon>0$
such that, for any
$t\geq t_{\mu}$
,
\begin{equation*} \sup_{|f|\leq\psi}|\mathbb{E}_{\mu}[\,f(X_{t})\,|\,T_{0}>t]-\alpha(f)|\leq \max\{C_{1}, C_{2}\}\bigg[\alpha\bigg(\frac{\psi^2}{\eta}\bigg)\bigg]^{\frac{1}{2}}\bigg\|\frac{\textrm{d}(\eta\circ\mu)}{\textrm{d}(\eta\circ\alpha)}-1\bigg\|_{2}\textrm{e}^{-\varepsilon t},\end{equation*}
where
\begin{eqnarray*} C_{1}=\left(1+\frac{1+\alpha(\psi)}{1-b}\right) , \qquad C_{2}=2+\alpha(\psi),\end{eqnarray*}
and b is a constant on
$(0, 1)$
.
The rest of this paper is organized as follows. In Section 2 we present some preliminaries that will be needed later. In Section 3 we give the proof of Theorem 1.1. The proof of Theorem 1.2 is given in Section 4.
2. Preliminaries
In this section we introduce the Q-process as an h-transform for the sub-Markovian semigroup
$(P_{t})_{t\geq0}$
, and preliminary facts which will be used later.
Let
$(Q_{i}(x),i\geq0)$
be a sequence of birth–death polynomials satisfying the recurrence relation
\begin{equation} \begin{aligned} & Q_{0}(x)=0, \qquad Q_{1}(x)=1, \\ & b_{i}Q_{i+1}(x)-(b_{i}+d_i)Q_{i}(x)+d_iQ_{i-1}(x)=-x Q_{i}(x), \qquad i\in\mathbb{N}. \end{aligned}\end{equation}
We write
$P_{ij}(t)=\mathbb{P}_{i}(X_{t}=j)$
. Under our assumptions, from [Reference Anderson1, Theorem 5.1.9] we know that there exists a parameter
$\lambda_{c}\geq0$
, called the decay parameter of the process X, such that, for all
$i,j\in\mathbb{N}$
,
$\lambda_{c}=-\lim_{t\to\infty}({1}/{t})\log P_{ij}(t)$
. To ensure the existence of quasi-stationary distributions, the decay parameter is usually required to be strictly greater than 0. On the existence and uniqueness of quasi-stationary distributions for birth–death processes, we have the following exact and detailed results.
Theorem 2.1. (van Doorn [Reference van Doorn16].) Let X be a birth–death process satisfying
$A=\infty$
.
-
(i) If
$\lambda_{c}=0$
, then there is no quasi-stationary distribution. -
(ii) If
$S=\infty$
and
$\lambda_{c}>0$
, then there is a one-parameter family of quasi-stationary distributions given by
$\alpha_{\lambda}(i)=({\pi_i}/{d_1})\lambda Q_{i}(\lambda)$
for
$\alpha_{\lambda}(i)$
,
$0<\lambda\leq\lambda_{c}$
,
$i\in\mathbb{N}$
. -
(iii) If
$S<\infty$
then
$\lambda_{c}>0$
and there is precisely one quasi-stationary distribution given by
\begin{equation*}\alpha=\bigg(\alpha_{\lambda_{c}}(i)=\frac{\pi_{i}\eta(i)}{\pi(\eta)}=\frac{\pi_i}{d_1}\lambda_{c}\eta(i), i\in\mathbb{N}\bigg).\end{equation*}
Remark 2.1. According to [Reference Chen5], we know that
$\lambda_{c}>0$
if and only if
\begin{equation*}\delta\,:\!=\,\sup\limits_{n\geq1}\sum_{i=1}^{n}\frac{1}{d_i\pi_i}\sum_{j=n}^{\infty}\pi_{j}<\infty.\end{equation*}
The process X conditioned to never be absorbed, usually referred to as the Q-process, defined by
$Y=(Y_{t},t\geq0)$
, plays a key role in the proofs of our main results. If
$\lambda_{c}>0$
, we know from [Reference Collet, Martnez and San Martn6, Proposition 5.9] that the Q-process Y, whose law starting from
$i\in\mathbb{N}$
is given by
$\mathbb{Q}_{i}(Y_{s_{1}}=i_{1},\ldots,Y_{s_{k}}=i_{k})=\lim_{t\rightarrow\infty}\mathbb{P}_{i}(X_{s_{1}}=i_{1},\ldots,X_{s_{k}}=i_{k} \mid T_{0}>t)$
, is a Markov chain taking values in
$\mathbb{N}$
, with transition kernel, for all
$i,j\in \mathbb{N}$
,
\begin{equation} \mathbb{Q}_{i}(Y_{s}=j)=\textrm{e}^{\lambda_{c}s}\frac{\eta(j)}{\eta(i)}\mathbb{P}_{i}(X_{s}=j).\end{equation}
Let
$(Q_{t})_{t\geq0}$
be the semigroup of the process Y under
$\mathbb{Q}$
. For all bounded and measurable functions f on
$\mathbb{N}$
and
$t\geq0$
, the equality (2.2) implies that, for all
$i\in \mathbb{N}$
,
\begin{equation}Q_{t}f(i)=\frac{\textrm{e}^{\lambda_{c}t}}{\eta(i)}P_{t}(\eta f)(i).\end{equation}
From (2.3), we have, for all
$i\in \mathbb{N}$
,
\begin{equation}P_{t}f(i)=\eta(i)\textrm{e}^{-\lambda_{c}t}Q_{t}\bigg(\frac{f}{\eta}\bigg)(i).\end{equation}
According to [Reference Collet, Martnez and San Martn6, Chapter 5], we know that the process Y is still a birth–death process taking values in
$\mathbb{N}$
with birth and death parameters given, for all
$i\in \mathbb{N}$
, by
\begin{equation*} \widetilde{{b}}_{i}=\frac{\eta(i+1)}{\eta(i)}{b}_{i}, \qquad \widetilde{{d}}_{i}=\frac{\eta(i-1)}{\eta(i)}{d}_{i}.\end{equation*}
We can compute the coefficients
$\widetilde{\pi}=(\widetilde{\pi}_{i},i\in \mathbb{N})$
analogous to (1.1):
\begin{equation}\widetilde{\pi}_{1}=1 , \qquad \widetilde{\pi}_{i}=\frac{\widetilde{{b}}_{1}\widetilde{{b}}_{2}\cdots\widetilde{{b}}_{i-1}}{\,\widetilde{{\!d}}_{2}\,\widetilde{{\!d}}_{3}\cdots\,\widetilde{{\!d}}_{i}}=\eta^{2}(i){\pi}_{i}, \qquad i\geq 2.\end{equation}
We define
$\widetilde{P}_{ij}(t)=\mathbb{Q}_{i}(Y_{t}=j)$
for all
$i,j\in\mathbb{N}$
. Then, by (2.2) and (2.5), we get
$\widetilde{\pi}_{i}\widetilde{P}_{ij}(t)=\widetilde{\pi}_{j}\widetilde{P}_{ji}(t)$
for all
$i,j\in \mathbb{N}$
. Namely, the process Y is reversible with respect to
$\widetilde\pi$
.
For the birth–death process X satisfying
$A=\infty$
and
$S<\infty$
, we know from [Reference Gao and Mao8, Reference He, Zhang and Zhu9] that
$\eta(i)$
is strictly increasing with
$i\in \mathbb{N}$
. When
$i\in \mathbb{N}$
, from (2.1), we see that
$\eta(i)$
has the minimum value 1. Furthermore, we also have the following result.
Proposition 2.1. ([Reference Gao and Mao8], Lemma 3.4.) Let X be a birth and death process satisfying
$A=\infty$
and
$S<\infty$
. Then
$\eta(\infty)\,:\!=\,\lim_{i\to\infty}\eta(i)<\infty$
.
Proposition 2.1 plays a key role in the proofs of our main results. From Proposition 2.1 we get
$\mu(\eta)<\infty$
, so (1.5) is well-defined.
We can see that one of the main features of the Q-process Y is that it is an h-transform of the original absorbed process X. The equality (2.3) naturally suggests the use of the h-transform to deduce quasi-stationary properties. This general method has been used successfully in, for example, [Reference Diaconis and Miclo7, Reference Oçafrain13, Reference Oçafrain14, Reference Takeda15]. Here, we also use the h-transform to study the quasi-stationarity of birth–death processes.
3. Proof of Theorem 1.1
We only need to show that (ii) and (iii) are equivalent. If (iii) holds, then there exists a unique quasi-stationary distribution and the distribution
$\alpha$
defined in Theorem 2.1 is the unique quasi-stationary distribution. That is, (ii) holds.
If (ii) holds then
$S<\infty$
, so we know from the proof of [Reference He, Zhang and Zhu9, Theorem 3.1] that the Q-process Y is strongly ergodic, which means that
$\lim_{t\rightarrow\infty}\sup_{i}\sum_{j\in\mathbb{N}}|\widetilde{P}_{ij}(t)-m_{j}|=0$
, where
$m=(m_{j}={\pi_{j}\eta^{2}(j)} / {\pi(\eta^{2})}, j\in\mathbb{N})$
is the unique stationary distribution of the process Y. It is well known (see, e.g., [Reference Anderson1]) that strong ergodicity implies exponential ergodicity. So, if the process Y is strongly ergodic, then there exist two constants
$C, \gamma>0$
such that, for any
$i\in\mathbb{N}$
,
\begin{equation}\left\|\mathbb{Q}_{i}(Y_t\in \cdot)-m\right\|_{\textrm{TV}}\leq C\textrm{e}^{-\gamma t}.\end{equation}
According to Proposition 2.1, we know that when
$i\in\mathbb{N}$
,
$1\leq\eta(i)\leq\eta(\infty)<\infty$
. Therefore, if f(i) is a bounded and measurable function on
$\mathbb{N}$
, then
${f(i)} / {\eta(i)}$
is also a bounded and measurable function on
$\mathbb{N}$
. Thus, from (2.4), for all
$t\geq0$
, all probability measure
$\mu$
on
$\mathbb{N}$
, and
$f\in\mathcal{B}_{1}(\mathbb{N})$
, we have
\begin{equation*} \mathbb{E}_{\mu}[f(X_{t}) \mid T_{0}>t] = \frac{\mu(P_{t}f)}{\mu(P_{t}\textbf{1})} = \frac{\textrm{e}^{-\lambda_{c}t}\mu\!\left(\eta Q_{t}({f} / {\eta})\right)}{\textrm{e}^{-\lambda_{c}t}\mu\!\left(\eta Q_{t}({\textbf{1}} / {\eta})\right)} = \frac{(\eta\circ\mu)Q_{t}({f} / {\eta})}{(\eta\circ\mu)Q_{t}({\textbf{1}} / {\eta})}.\end{equation*}
Note that
\begin{equation*} m(\,{f} / {\eta})=\alpha(f)\frac{\pi(\eta)}{\pi(\eta^{2})}=\frac{\alpha(f)}{\alpha(\eta)}\leq\alpha(f).\end{equation*}
So, for any
$f\in\mathcal{B}_{1}(\mathbb{N})$
, by (3.1) we get
\begin{equation} \begin{aligned} & |(\eta\circ\mu)Q_{t}({f} / {\eta})-\alpha(f)| \leq |(\eta\circ\mu)Q_{t}({f} / {\eta})-m({f} / {\eta})|\leq C\textrm{e}^{-\gamma t} , \\ & |(\eta\circ\mu)Q_{t}({\textbf{1}} / {\eta})-1| \leq C\textrm{e}^{-\gamma t} . \end{aligned}\end{equation}
Therefore, combining the inequalities in (3.2), for any
$t>({\log C}) / {\gamma}$
we have
\begin{equation*} \frac{\alpha(f)-C\textrm{e}^{-\gamma t}}{1+C\textrm{e}^{-\gamma t}}\leq\mathbb{E}_{\mu}[f(X_{t}) \mid T_{0}>t]\leq\frac{\alpha(f)+C\textrm{e}^{-\gamma t}}{1-C\textrm{e}^{-\gamma t}}.\end{equation*}
From (3.2) we have relations of the type
$\mathbb{E}_{\mu}[f(X_{t}) \mid T_{0}>t]={a(t)} / {b(t)}$
, with
$a(t)=\alpha(f)+\delta(t)$
,
$b(t)=1+\epsilon(t)$
, and
$\max\{|\delta(t)|,|\epsilon(t)|\}\leq C\textrm{e}^{-\gamma t}$
. Then, it suffices to use the expansion
\begin{equation*} \frac{1}{1+\epsilon(t)}=1-\epsilon(t)+\frac{\epsilon^{2}(t)}{1+\epsilon(t)}\end{equation*}
to get that
$|\mathbb{E}_{\mu}[f(X_{t}) \mid T_{0}>t]-\alpha(f)|$
is bounded by
$C^{\prime}\textrm{e}^{-\gamma t}$
for some finite constant C
′, and the result follows straightforwardly.
4. Proof of Theorem 1.2
In this section we give the proof of Theorem 1.2, which is similar to [Reference Oçafrain14, Theorem 2.1] where the author considered the exponential convergence of conditioned distributions to a quasi-stationary distribution in total variation and in 1-Wasserstein distance for general Markov processes under several difficult-to-check conditions. For birth–death processes we have a much simpler and explicit condition. Our more restricted context enables us to obtain a more detailed result.
We only consider initial measures
$\mu$
on
$\mathbb{N}$
such that
$\|({\textrm{d}(\eta\circ\mu)} / {\textrm{d} m})-1\|_{2}<+\infty$
, since if
$\|({\textrm{d}(\eta\circ\mu)} / {\textrm{d} m})-1\|_{2}=+\infty$
then Theorem 1.2 is trivially satisfied. Recall that if the birth–death process X satisfies
$A=\infty$
and
$S<\infty$
, then the Q-process Y is strongly ergodic. Thus, we know from [Reference Chen4, Theorem 1.1] that
$(Q_{t})_{t\geq0}$
converges exponentially in the
$\mathbb{L}^2(m)$
-norm, i.e. there is a positive
$\varepsilon$
such that, for all
$f\in\mathbb{L}^2(m)$
and
$t\geq0$
,
\begin{equation}\|Q_{t}f-m(f)\|_{2}\leq\|f-m(f)\|_{2}\textrm{e}^{-\varepsilon t}.\end{equation}
Note that
$\eta$
is bounded on
$\mathbb{N}$
and has the minimum value 1, so if f is a measurable function on
$\mathbb{N}$
such that
$|f|\leq\psi$
and
$\alpha(\psi^{2})<+\infty$
, then
${f} / {\eta}$
is also a measurable function on
$\mathbb{N}$
and belongs to
$\mathbb{L}^2(m)$
. From Section 2, we know that the process Y is reversible with respect to
$\widetilde\pi$
, which implies reversibility with respect to m. Thus, by (4.1) and the Cauchy–Schwarz inequality, for any probability measure
$\mu$
on
$\mathbb{N}$
and any measurable function f on
$\mathbb{N}$
such that
$|f|\leq\psi$
, we have
\begin{align*} \sup\limits_{|f|\leq\psi}\bigg|\mu Q_{t}\bigg(\frac{f}{\eta}\bigg)-\alpha(f)\bigg| & \leq \sup\limits_{|f|\leq\psi}\bigg|\mu Q_{t}\bigg(\frac{f}{\eta}\bigg)-m\bigg(\frac{f}{\eta}\bigg)\bigg| \\ & = \sup\limits_{|f|\leq\psi}\bigg|m\bigg(\frac{\textrm{d} \mu}{\textrm{d} m}Q_{t}\bigg(\frac{f}{\eta}\bigg)-\frac{f}{\eta}\bigg)\bigg| \\ & = \sup\limits_{|f|\leq\psi}\bigg|m\bigg(\frac{f}{\eta}Q_{t}\bigg(\frac{\textrm{d}\mu}{\textrm{d} m}\bigg)-\frac{f}{\eta}\bigg)\bigg| \\ & = \sup\limits_{|f|\leq\psi}\bigg|m\bigg[\frac{f}{\eta}\left(Q_{t}\bigg(\frac{\textrm{d}\mu}{\textrm{d} m}-1\bigg)\right)\bigg]\bigg| \\ & \leq \bigg[m\bigg(\frac{\psi^2}{\eta^2}\bigg)\bigg]^{\frac{1}{2}}\bigg\|\frac{\textrm{d}\mu}{\textrm{d} m}-1\bigg\|_{2}\textrm{e}^{-\varepsilon t} \\ & \leq \bigg[\alpha\bigg(\frac{\psi^2}{\eta}\bigg)\bigg]^{\frac{1}{2}}\bigg\|\frac{\textrm{d}\mu}{\textrm{d} m}-1\bigg\|_{2}\textrm{e}^{-\varepsilon t}.\end{align*}
Note that
\begin{eqnarray*} \mathbb{E}_{\mu}[f(X_{t}) \mid T_{0}>t] = \frac{(\eta\circ\mu)Q_{t}({f} / {\eta})}{(\eta\circ\mu)Q_{t}({\textbf{1}} / {\eta})},\end{eqnarray*}
so, for any
$t>\{\log\![(\alpha({\psi^2} / {\eta}))^{{1} / {2}}\|({\textrm{d}(\eta\circ\mu)} / {\textrm{d} m})-1\|_{2}]\} / {\varepsilon}$
, we get
\begin{align} \frac{\alpha(f)-(\alpha({\psi^2} / {\eta}))^{{1} / {2}}\|({\textrm{d}(\eta\circ\mu)} / {\textrm{d} m})-1\|_{2}\textrm{e}^{-\varepsilon t}}{1+(\alpha({\psi^2} / {\eta}))^{{1} / {2}}\|({\textrm{d}(\eta\circ\mu)} / {\textrm{d} m})-1\|_{2}\textrm{e}^{-\varepsilon t}} \\ \cr&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \leq\mathbb{E}_{\mu}[f(X_{t}) \mid T_{0}>t] \leq\frac{\alpha(f)+(\alpha({\psi^2} / {\eta}))^{{1} / {2}}\|({\textrm{d}(\eta\circ\mu)} / {\textrm{d} m})-1\|_{2}\textrm{e}^{-\varepsilon t}}{1-(\alpha({\psi^2} / {\eta}))^{{1} / {2}}\|({\textrm{d}(\eta\circ\mu)}{\textrm{d} m})-1\|_{2}\textrm{e}^{-\varepsilon t}}.\end{align}
Since
$\alpha(\psi^{2})<+\infty$
, by the Cauchy–Schwarz inequality we have
$\alpha(\psi)<+\infty$
. Thus, by (4.2), for any
$t>\{\log\!((\alpha({\psi^2} / {\eta}))^{{1} / {2}}\|({\textrm{d}(\eta\circ\mu)} / {\textrm{d} m}) - 1\|_{2})\} / {\varepsilon}$
, we obtain
\begin{eqnarray*} \sup\limits_{|f|\leq\psi}|\mathbb{E}_{\mu}[f(X_{t}) \mid T_{0}>t]-\alpha(f)|\leq \max\{C_{1}, C_{2}\}\bigg(\alpha\bigg(\frac{\psi^2}{\eta}\bigg)\bigg)^{\frac{1}{2}}\bigg\|\frac{\textrm{d}(\eta\circ\mu)}{\textrm{d} m}-1\bigg\|_{2}\textrm{e}^{-\varepsilon t},\end{eqnarray*}
where
\begin{eqnarray*}C_{1}\,:\!=\,\left(1+\frac{1+\alpha(\psi)}{1-b}\right) , \qquad C_{2}\,:\!=\,2+\alpha(\psi),\end{eqnarray*}
and b is a constant on (0, 1).
Set
$\phi_{t}(\mu)\,:\!=\,\mathbb{P}_{\mu}(X_{t}\in\cdot|T_{0}>t)$
. For any
$t\geq0$
and any probability measure
$\mu$
on
$\mathbb{N}$
, we know from [Reference Oçafrain14, Lemma 2.7] that
\begin{equation}\eta\circ\phi_{t}(\mu)=(\eta\circ\mu)Q_{t}.\end{equation}
There exists
$t_{\mu}\geq0$
such that, for any
$t\geq t_{\mu}$
,
\begin{equation*}\bigg(\alpha\bigg(\frac{\psi^2}{\eta}\bigg)\bigg)^{\frac{1}{2}}\bigg\|\frac{\textrm{d}(\eta\circ\phi_{t}(\mu))}{\textrm{d} m}-1\bigg\|_{2}\textrm{e}^{-\varepsilon t}\lt b.\end{equation*}
Hence, by (4.1), (4.3), and the above result, for any
$t\geq t_{\mu}$
, we get
\begin{align*} \sup\limits_{|f|\leq\psi}|\mathbb{E}_{\mu}[f(X_{t}) \mid T_{0}>t]-\alpha(f)| & \leq \max\{C_{1}, C_{2}\}\bigg(\alpha\bigg(\frac{\psi^2}{\eta}\bigg)\bigg)^{\frac{1}{2}}\bigg\|\frac{\textrm{d}(\eta\circ\phi_{t_{\mu}}(\mu))}{\textrm{d} m}-1\bigg\|_{2}\textrm{e}^{-\varepsilon (t-t_{\mu})} \\ & \leq \max\{C_{1}, C_{2}\}\bigg(\alpha\bigg(\frac{\psi^2}{\eta}\bigg)\bigg)^{\frac{1}{2}}\bigg\|\frac{\textrm{d}(\eta\circ\mu)}{\textrm{d} m}-1\bigg\|_{2}\textrm{e}^{-\varepsilon t}.\end{align*}
This ends the proof of Theorem 1.2.
Acknowledgements
The authors would like to thank two anonymous referees for carefully reading this paper, and for various helpful remarks and suggestions.
Funding information
This work was supported by the National Natural Science Foundation of China (Grant No. 12001184) and Outstanding Youth Project of Education Department of Hunan Province (Grant No. 19B307).
Competing interests
There were no competing interests to declare which arose during the preparation or publication process of this article.
