2. Weak dependence and strong mixing conditions

We assume that all random variables and processes are defined on a given probability space $(\Omega, \mathcal{A},\mathbb{P})$ .

We let $\mathbb{N}^*$ refer to the set of positive integers, $\mathbb{N}$ the set of the non-negative integers, $\mathbb{Z}$ the set of all integers, and $\mathbb{R}_+$ the set of non-negative real numbers. We denote the Euclidean norm by $\| \!\cdot\! \|$ . However, due to the equivalence of all norms, none of our results depend on the chosen norm.

Although the theory developed below is most relevant for sampling processes defined in continuous time, we work with a general index set $\mathcal{I}$ , as this makes no difference and also covers other cases, such as a sampling of discrete-time processes or random fields.

Even if our theory extends to random fields, we always refer to X as a process for simplicity. Moreover, we consider

(2.1) \begin{equation}\mathcal{F}=\bigcup_{u \in \mathbb{N}} \mathcal{F}_u \quad \textrm{and} \quad \mathcal{G}=\bigcup_{v \in \mathbb{N}} \mathcal{G}_v,\end{equation}

where $\mathcal{F}_u$ and $\mathcal{G}_v$ , respectively, are two classes of measurable functions from $(\mathbb{R}^d)^u$ to $\mathbb{R}$ and $(\mathbb{R}^d)^v$ to $\mathbb{R}$ that we specify individually later on. Finally, for a function that is unbounded or not Lipschitz, we set its $\|\!\cdot\!\|_{\infty}$ norm or Lipschitz constant equal to infinity.

Definition 2.2. Let $\mathcal{I}$ be an index set as in Definition 2.1, let $X=(X_t)_{t\in \mathcal{I}}$ be a process with values in $\mathbb{R}^d$ , and let $\Psi$ be a function from $\overline{\mathbb{R}_+^6}$ to $\mathbb{R}_+$ non-decreasing in all arguments. The process X is called $\Psi$ -weakly dependent if there exists a sequence of coefficients $\epsilon=(\epsilon(r))_{r \in \mathbb{R}_+}$ converging to 0 and satisfying inequality (1.1) for all

\begin{equation*}\begin{cases}(u,v) \in \mathbb{N}^* \times \mathbb{N}^*,\\[5pt]r\in \mathbb{R}_+, \\[5pt]\text{$I_u=\{i_1,\ldots,i_u\} \subseteq{\mathcal{I}} $ and $ J_v=\{\,j_1,\ldots,j_v\} \subseteq \mathcal{I}$, such that $d(I_u,J_v) \geq r$,}\\[5pt]\textrm{functions $ F \colon (\mathbb{R}^{d})^u \to \mathbb{R} $ and $ G\colon (\mathbb{R}^{d})^v \to \mathbb{R} $ belonging to $\mathcal{F}$ and $\mathcal{G}$ respectively,}\end{cases}\end{equation*}

where c is a constant independent of r.

Without loss of generality we always consider $\epsilon$ to be a non-increasing sequence of coefficients.

The first covariance inequality for Lipschitz functions of positively or negatively associated random fields in the literature appears in [Reference Bulinski and Shabanovich9]. Since this result, other covariance inequalities have been determined for functions F and G being either bounded Lipschitz or bounded measurable functions of processes and random fields. In the latter set-up, Definition 2.2 encompasses the so-called weak dependence conditions as described in [Reference Bulinski and Shashkin11, Definition 5.12] and [Reference Dedecker, Doukhan, Lang, León, Louhichi and Prieur20, Definition 2.2] for $\mathcal{I}=\mathbb{Z}, \mathbb{Z}^m$ . Therefore several sequences of coefficients $\epsilon$ satisfying Definition 2.2 are already well known.

• Let $\mathcal{F}=\mathcal{G}$ and $\mathcal{F}_u$ be the class of bounded Lipschitz functions from $(\mathbb{R}^d)^u$ to $\mathbb{R}$ with respect to the distance $\delta$ on $(\mathbb{R}^d)^u$ defined by

  (2.2) \begin{equation}\delta(x^*,y^*)= \sum_{i=1}^u \|x_i-y_i\|,\end{equation}
  where $x^*=(x_1,\ldots,x_u)$ and $y^*=(y_1,\ldots,y_u)$ , and $x_i,y_i \in \mathbb{R}^d$ for all $i=1,\ldots,u$ . Then
  \begin{equation*}\mathrm{Lip}(F)=\sup_{x\neq y} \dfrac{|F(x)-F(y)|}{\| x_1-y_1 \|+\|x_2-y_2\|+ \ldots+ \|x_d-y_d\|}.\end{equation*}
  For
  \begin{equation*}\Psi(\|F\|_{\infty},\|G\|_{\infty},\mathrm{Lip}(F),\mathrm{Lip}(G),u,v)=u\;\mathrm{Lip}(F) \|G\|_{\infty} +v\;\mathrm{Lip}(G) \|F\|_{\infty},\end{equation*}
  $\epsilon$ corresponds to the $\eta$ -coefficients as defined in [Reference Doukhan and Louhichi25]. An extension of this definition for $\mathcal{I}=\mathbb{Z}^m$ is given in [Reference Doukhan and Lang24]. If, instead,
  \begin{align*}&\Psi(\|F\|_{\infty},\|G\|_{\infty},\mathrm{Lip}(F),\mathrm{Lip}(G),u,v)\\[5pt]&\quad = u\;\mathrm{Lip}(F) \|G\|_{\infty} +v\;\mathrm{Lip}(G) \|F\|_{\infty} +uv\;\mathrm{Lip}(F)\;\mathrm{Lip}(G) \nonumber,\end{align*}
  then $\epsilon$ corresponds to the $\lambda$ -coefficients as defined in [Reference Doukhan and Wintenberger26] for $\mathcal{I}=\mathbb{Z}$ and in [Reference Dedecker, Doukhan, Lang, León, Louhichi and Prieur20, Remark 2.1] for $\mathcal{I}=\mathbb{Z}^m$ . Moreover, for
  \begin{equation*}\Psi(\|F\|_{\infty},\|G\|_{\infty},\mathrm{Lip}(F),\mathrm{Lip}(G),u,v)= uv\;\mathrm{Lip}(F)\;\mathrm{Lip}(G),\end{equation*}
  $\epsilon$ corresponds to the $\kappa$ -coefficients, and, for
  \begin{equation*}\Psi(\|F\|_{\infty},\|G\|_{\infty},\mathrm{Lip}(F),\mathrm{Lip}(G),u,v)=\min\!(u,v) \mathrm{Lip}(F)\;\mathrm{Lip}(G),\end{equation*}
  $\epsilon$ corresponds to the $\zeta$ -coefficients as defined in [Reference Doukhan and Louhichi25]. The definition of $\zeta$ -weak dependence for $\mathcal{I}=\mathbb{Z}^m$ can be found in [Reference Bulinski and Suquet12].

• Let $\mathcal{F}_u$ be the class of bounded measurable functions from $(\mathbb{R}^d)^u$ to $\mathbb{R}$ and let $\mathcal{G}_v$ be the class of bounded Lipschitz functions from $(\mathbb{R}^d)^v$ to $\mathbb{R}$ with respect to the distance $\delta$ defined in (2.2). Then, for

  \begin{equation*}\Psi(\|F\|_{\infty},\|G\|_{\infty},\mathrm{Lip}(F),\mathrm{Lip}(G),u,v)= v \| F\|_{\infty} \;\mathrm{Lip}(G),\end{equation*}
  $\epsilon$ corresponds to the $\theta$ -coefficients as defined in [Reference Dedecker, Doukhan, Lang, León, Louhichi and Prieur20]. An extension of this definition for $\mathcal{I}=\mathbb{Z}^m$ appears in [Reference Dedecker, Doukhan, Lang, León, Louhichi and Prieur20, Remark 2.1]. Moreover, an alternative definition for this notion of dependence is given in [Reference Dedecker and Doukhan19] for $\mathcal{F}_u$ as above and $\mathcal{G}_1$ the class of Lipschitz function from $\mathbb{R}^d$ to $\mathbb{R}$ , for $\mathcal{I}=\mathbb{Z}$ .

The extension to index sets $\mathcal{I}=\mathbb{R},\mathbb{R}^m$ of the weak dependence notions described above is straightforward.

We now show that Definition 2.2 also encompasses strong mixing. We first define the strong mixing coefficient [Reference Rosenblatt39].

We suppose that $\mathcal{A}_1$ and $\mathcal{A}_2$ are sub- $\sigma$ -fields of $\mathcal{A}$ and define

\begin{equation*}\alpha(\mathcal{A}_1, \mathcal{A}_2)\;:\!=\;\sup_{\substack{ A \in \mathcal{A}_1\\ B \in \mathcal{A}_2}}|P(A \cap B) - P(A)P(B)|.\end{equation*}

Let $\mathcal{I}$ be a set as in Definition 2.1. Then a process $X=(X_t)_{t \in \mathcal{I}}$ with values in $\mathbb{R}^d$ is said to be $\alpha_{u,v}$ -mixing for $u,v\in\mathbb{N}\cup\{\infty\}$ if

(2.3) \begin{equation}\alpha_{u,v}(r)\;:\!=\; \sup \{ \alpha(\mathcal{A}_{\Gamma_1}, \mathcal{B}_{\Gamma_2})\colon \Gamma_1, \Gamma_2\subseteq \mathcal{I}, |\Gamma_1| \leq u, |\Gamma_2| \leq v, d(\Gamma_1,\Gamma_2) \geq r \}\end{equation}

converges to zero as $r \to \infty$ , where $\mathcal{A}_{\Gamma_1}=\sigma( X_i \colon i \in \Gamma_1)$ and $\mathcal{B}_{\Gamma_2}=\sigma( X_j\colon j \in \Gamma_2)$ . If we let $\alpha(r)=\alpha_{\infty,\infty}(r)$ , it is apparent that $\alpha_{u,v}(r) \leq \alpha(r)$ . If $\alpha(r) \to 0$ as $r \to \infty$ , then X is simply said to be $\alpha$ -mixing. For a comprehensive discussion of the coefficients $\alpha_{u,v}(r)$ , $\alpha(r)$ and their relation to other strong mixing coefficients, we refer to [Reference Bradley6], [Reference Bulinski8], and [Reference Dedecker18].

Proposition 2.1. Let $\mathcal{I}$ be a set as in Definition 2.1 and let $X=(X_t)_{t\in \mathcal{I}}$ be a process with values in $\mathbb{R}^d$ and $\mathcal{F}=\mathcal{G}$ , where $\mathcal{F}_u$ is the class of bounded measurable functions from $(\mathbb{R}^d)^u$ to $\mathbb{R}$ . X is $\alpha$ -mixing if and only if there exists a sequence $(\epsilon(r))_{r \in \mathbb{R}_+}$ converging to 0 such that inequality (1.1) holds with

(2.4) \begin{equation}\Psi(\|F\|_{\infty},\|G\|_{\infty},\mathrm{Lip}(F),\mathrm{Lip}(G),u,v)=\|F\|_{\infty}\|G\|_{\infty}.\end{equation}

Proof. Set $\mathcal{A}_{I_u}=\sigma(X_i\colon i \in I_u)$ and $\mathcal{B}_{J_v}=\sigma(X_j\colon j \in I_v)$ . For arbitrary $(u,v) \in \mathbb{N}^* \times \mathbb{N}^*$ and $r \in \mathbb{R}_+$ , let $I_u=\{i_1,\ldots,i_u\}$ and $J_v=\{j_1,\ldots,j_v\}$ be arbitrary subsets of $\mathcal{I}$ such that $d(I_u,J_v) \geq r$ . Moreover, choose arbitrary $F \in \mathcal{F}_u$ and $G \in \mathcal{G}_v$ . By [Reference Ibragimov and Linnik30, Theorem 17.2.1],

\begin{equation*}|\mathrm{Cov}(F(X_{i_1},\ldots,X_{i_u}),G(X_{j_1},\ldots,X_{j_v}))| \leq 4 \, \alpha(\mathcal{A}_{I_u},\mathcal{B}_{I_v}) \, \|F\|_{\infty}\|G\|_{\infty}.\end{equation*}

Definition (2.3) immediately implies that the right-hand side of the inequality above is smaller than or equal to $4 \alpha(r) \,\|F\|_{\infty} \,\|G\|_{\infty}$ . Hence, if X is $\alpha$ -mixing then (1.1) holds with $\epsilon(r)=\alpha(r)$ and $c=4$ .

We assume now that the sequence X is $\Psi$ -weakly dependent with $\Psi$ given by (2.4). By [Reference Ibragimov and Linnik30, Theorem 17.2.1] and [Reference Bradley6, Remark 3.17(ii)], we can rewrite the definition of the $\alpha$ -coefficients as

\begin{align*}\alpha(r)&= \sup_{\substack{\Gamma_1, \Gamma_2 \subseteq \mathcal{I}\\|\Gamma_1|< \infty, |\Gamma_2|<\infty\\ d(\Gamma_1,\Gamma_2) \geq r}}\alpha(\mathcal{A}_{\Gamma_1},\mathcal{A}_{\Gamma_2}) \notag \\[5pt] &=\sup_{(u,v) \in \mathbb{N}\times \mathbb{N}} \sup_{\substack{I_u, J_v \subseteq \mathcal{I}\\ d(I_u,J_v) \geq r } } \sup_{\substack{F \in \mathcal{F}_u\\ G \in \mathcal{G}_v }} \biggl\{ \dfrac{1}{4 \|F\|_{\infty} \|G\|_{\infty}} |\mathrm{Cov}(F(X_{i_1},\ldots,X_{i_u}),G(X_{j_1},\ldots,X_{j_v}))| \biggr\}.\end{align*}

Hence

\begin{equation*}\alpha(r) \leq \dfrac{c}{4} \epsilon(r).\end{equation*}

If X is $\Psi$ -weakly dependent, then X is $\alpha$ -mixing.

• Let $\mathcal{F}_u$ be the class of bounded measurable functions from $(\mathbb{R}^d)^u$ to $\mathbb{R}$ and let $\mathcal{G}_1$ be the class of bounded Lipschitz functions from $\mathbb{R}^d$ to $\mathbb{R}$ with respect to the distance $\delta$ defined in (2.2). Moreover, let $I_u=\{i_1, \ldots, i_u\} \subset \mathbb{R}^m$ , and let $j \in \mathbb{R}^m$ be such that $i_s<_{\mathrm{lex}} j$ for all $s=1,\ldots,u$ , and $\mathrm{dist}(I_u,j)\geq r$ . Then inequality (1.1) holds for $\Psi(\|F\|_{\infty},\|G\|_{\infty},\mathrm{Lip}(F),\mathrm{Lip}(G),u,1)= \| F\|_{\infty}\;\mathrm{Lip}(G),$ and $\epsilon$ corresponds to the $\theta$ -lex-coefficients.

For $\mathcal{I}=\mathbb{Z}^m$ , this notion of dependence is more general than $\alpha_{\infty,1}$ -mixing as defined in (2.3), that is, it applies to a broader class of models. Further, for $\mathcal{I}=\mathbb{Z}$ , $\theta$ -lex weak dependence is more general than the notion of $\alpha$ -mixing. We refer the reader to [Reference Curato, Stelzer and Ströh17, Section 2] for a complete introduction to $\theta$ -lex weak dependence and its properties.

3. Strong mixing and weak dependence coefficients under renewal sampling

Let X be a strictly stationary $\mathbb{R}^d$ -valued process, that is, for all $n \in \mathbb{N}$ and all $t_1,\ldots,t_n \in \mathcal{I}$ it holds that the finite-dimensional distributions (indicated by $\mathcal{L}(\!\cdot\!)$ ) are shift-invariant:

\begin{equation*}\mathcal{L}(X_{t_1+h},\ldots,X_{t_n+h})=\mathcal{L}(X_{t_1},\ldots,X_{t_n})\quad\text{for all $h \in \mathcal{I}$.}\end{equation*}

We want to investigate the asymptotic dependence of X sampled at a renewal sequence.

We use a definition of renewal process based on the sequence $\tau$ (see [Reference Hunter29]) and that agrees with the definition of a two-sided Lévy process (see [Reference Applebaum3, page 124]) in the Poisson case. Similar sampling schemes are used in [Reference Aït-Sahalia and Mykland1], [Reference Charlot and Rachdi14], and [Reference Lii and Masry33], for example.

Definition 3.1. Let $\mathcal{I} \subseteq \mathbb{R}^m$ be a set as in Definition 2.1 and let $\tau=(\tau_i)_{i \in \mathbb{Z}\setminus \{0\}}$ be an $\mathcal{I}$ -valued sequence of non-negative (component-wise) independent and identically distributed (i.i.d.) random vectors with distribution function $\mu$ such that $\mu\{0\} <1$ . For $i \in \mathbb{Z}$ , we define an $\mathcal{I}$ -valued stochastic process $(T_i)_{i \in \mathbb{Z}}$ as

\begin{align*} T_0\;:\!=\; 0 \quad \text{and} \quad T_i \;:\!=\; \begin{cases}\sum_{j=1}^i \tau_j , &i \in \mathbb{N} , \\[5pt] -\sum_{j=i}^{-1} \tau_j , &-i \in \mathbb{N}.\end{cases}\end{align*}

The sequence $(T_i)_{i \in \mathbb{Z}}$ is called a renewal sampling sequence. When $\mathcal{I} \subset \mathbb{R}$ , we let $\tau$ denote the sequence of inter-arrival times.

Definition 3.2. Let $X=(X_t)_{t \in \mathcal{I}}$ be a process with values in $\mathbb{R}^d$ and let $(T_i)_{i \in \mathbb{Z}}$ be a renewal sampling sequence independent of X. We define the sequence $Y=(Y_i)_{i\in \mathbb{Z}}$ as the stochastic process with values in $\mathbb{R}^{d+1}$ given by

\begin{equation*} Y_i=( X_{T_i}, T_i-T_{i-1} )^{\top}.\end{equation*}

We call X the underlying process and Y the renewal sampled process.

• Modern health monitoring systems such as smartphones or wearable devices such as smartwatches enable monitoring of the health conditions of patients by measuring heart rate, electrocardiogram, and body temperature, among other information; see [Reference Vitabile, Marks, Stojanovic, Pllana, Molina, Krzyszton, Sikora, Jarynowski, Hosseinpour, Jakobik, Illic, Respicio, Moldovan, Pop and Salomie43]. These measurements are records on a discrete-time grid, mostly irregularly distributed. In these cases, observation times depend on the measuring instrument (typically sensors), i.e. on a random source independent of the process X, as observed in [Reference Bardet and Bertrand4]. In this context, the hypothesis of independence of T and X is entirely realistic.

• Measurements from spatio-temporal random fields such as temperature, vegetation, or population are now recorded over a set of moving or fixed locations in space and time, typically not regularly distributed. These data sets are called point reference or raster data and are analyzed in earth science, for example. Further, GPS data, e.g. of a taxi, which periodically transmit the location of an object over time, are an example of spatio-temporal data sets called trajectory data, which are typically irregularly distributed in space and time. We refer the reader to the survey [Reference Wang, Cao and Yu44] and the references therein for an account of the data sets above and their practical relevance. The hypothesis of independence of T and X seems realistic for these data because their sampling in space–time depends on the instrument used to record them.

In the following theorem, we work with the class of functions defined in (2.1) and

\begin{equation*} \mathcal{\tilde{F}}=\bigcup_{u \in \mathbb{N}} \mathcal{\tilde{F}}_u \quad \text{and} \quad \mathcal{\tilde{G}}=\bigcup_{v \in \mathbb{N}} \mathcal{\tilde{G}}_v,\end{equation*}

where $\mathcal{\tilde{F}}_u$ and $\mathcal{\tilde{G}}_v$ are respectively two classes of measurable functions from $(\mathbb{R}^{d+1})^u$ to $\mathbb{R}$ and $(\mathbb{R}^{d+1})^v$ to $\mathbb{R}$ which can be either bounded or bounded Lipschitz.

Theorem 3.1. Let $Y=(Y_i)_{i \in \mathbb{Z}}$ be a renewal sampled process with the underlying process X being strictly stationary and $\Psi$ -weakly dependent with coefficients $\epsilon$ . Then Y is a strictly stationary process, and there exists a sequence $\mathcal{E}$ such that

\begin{equation*} |\mathrm{Cov}(\tilde{F}(Y_{i_1},\ldots,Y_{i_u}),\tilde{G}(Y_{j_1},\ldots,Y_{j_v}))| \leq C \, \Psi(\|\tilde{F}\|_{\infty},\|\tilde{G}\|_{\infty},\mathrm{Lip}(\tilde{F}),\mathrm{Lip}(\tilde{G}),u,v) \, \mathcal{E}(n)\end{equation*}

for all

\begin{equation*}\begin{cases}(u,v) \in \mathbb{N}^* \times \mathbb{N}^*,\\[5pt] n\in \mathbb{N}, \\[5pt] \textit{$\{i_1,\ldots,i_u\} \subseteq \mathbb{Z} $ and $ \{j_1,\ldots,j_v\} \subseteq \mathbb{Z}$,} \\[5pt] \textit{with $ i_1\leq \ldots \leq i_u < i_u+n\leq j_1\leq \ldots\leq j_v$,} \\[5pt] \textit{functions $ \tilde{F} \colon (\mathbb{R}^{d+1})^u \to \mathbb{R} $ and $ \tilde{G}\colon (\mathbb{R}^{d+1})^v \to \mathbb{R} $ belonging to $\mathcal{\tilde{F}}$ and $\mathcal{\tilde{G}}$,}\end{cases}\end{equation*}

where C is a constant independent of n. Moreover,

(3.1) \begin{equation}\mathcal{E}(n)=\int_{\mathcal{I}} \epsilon(\|r\|) \, \mu^{*n}({\mathrm{d}} r),\end{equation}

where $\mu^{*0}$ is the Dirac delta measure in zero, and $\mu^{*n}$ is the n-fold convolution of $\mu$ for $n \geq 1$ .

Proof of Theorem 3.1. Y is a strictly stationary process by [Reference Brandes and Curato7, Proposition 2.1]. Consider arbitrary fixed $(u,v) \in \mathbb{N}^* \times \mathbb{N}^*$ , $n \in \mathbb{N}$ , $\{i_1,\ldots,i_u\} \subseteq \mathbb{Z}$ and $\{j_1,\ldots,j_v\} \subseteq \mathbb{Z}$ with $i_1\leq\ldots\leq i_u \leq i_u+n \leq j_1\leq \ldots\leq j_v$ , and functions $\tilde{F} \in \tilde{\mathcal{F}}$ and $\tilde{G} \in \tilde{\mathcal{G}}$ . Without loss of generality, let us consider throughout that $i_1 >0$ . Then, by conditioning with respect to the sequence $\tau$ and using the law of total covariance (see [Reference Chan, Guo, Lee and Li13, Proposition A.1]), we obtain

(3.2) \begin{align} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! |\mathrm{Cov}(\tilde{F}(Y_{i_1},\ldots,Y_{i_u}),\tilde{G}(Y_{j_1},\ldots,Y_{j_v})) | \nonumber \\[5pt] &\quad \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \leq | \mathbb{E}(\mathrm{Cov}(\tilde{F}(Y_{i_1},\ldots,Y_{i_u}), \tilde{G}(Y_{j_1},\ldots,Y_{j_v})\mid \tau_i \colon i=1,\ldots,j_v ))| \end{align}

(3.3) \begin{align} \quad\quad +\, | \mathrm{Cov}(\mathbb{E}(\tilde{F}(Y_{i_1},\ldots,Y_{i_u})\mid \tau_i\colon i=1,\ldots,j_v), \mathbb{E}(\tilde{G}(Y_{j_1},\ldots,Y_{j_v})\mid \tau_i \colon i=1,\ldots,j_v ) ) |. \end{align}

Let us first discuss the summand (3.3). We have

\begin{equation*}\mathbb{E}(\tilde{F}(Y_{i_1},\ldots,Y_{i_u})\mid \tau_i\colon i=1,\ldots,j_v)= \mathbb{E}(\tilde{F}(Y_{i_1},\ldots,Y_{i_u})\mid \tau_i\colon i=1,\ldots,i_u)\end{equation*}

because $\tilde{F}(Y_{i_1},\ldots,Y_{i_u})$ is independent of $\{\tau_i \colon i=i_{u}+1,\ldots,j_v\}$ . On the other hand,

\begin{align*}&\mathbb{E}(\tilde{G}(Y_{j_1},\ldots,Y_{j_v})\mid \tau_i \colon i=1,\ldots,j_v )\\[5pt] &\quad =\mathbb{E}\Bigl(\tilde{G}\Bigl(\Bigl(X_{T_{i_u}+\sum_{i=i_u+1}^{j_1} \tau_i}, \tau_{j_1}\Bigr)^{\prime},\ldots,\Bigl(X_{T_{i_u}+\sum_{i=i_u+1}^{j_v} \tau_i}, \tau_{j_v}\Bigr)^{\prime}\Bigr)\mid \tau_i \colon i=1,\ldots,j_v \Bigr),\end{align*}

and, by stationarity of the process X and the i.i.d. property of $(\tau_i)_{i \in \mathbb{Z} \setminus \{0\}}$ , it is equal to

\begin{align*}&\mathbb{E}\Bigl(\tilde{G}\Bigl(\Bigl(X_{\sum_{i=i_u+1}^{j_1} \tau_i}, \tau_{j_1}\Bigr)^{\prime},\ldots,\Bigl(X_{\sum_{i=i_u+1}^{j_v} \tau_i}, \tau_{j_v}\Bigr)^{\prime}\Bigr)\mid \tau_i \colon i=1,\ldots,j_v \Bigr) \\[5pt] &\quad = \mathbb{E}\Bigl(\tilde{G}\Bigl(\Bigl(X_{\sum_{i=i_u+1}^{j_1} \tau_i}, \tau_{j_1}\Bigr)^{\prime},\ldots,\Bigl(X_{\sum_{i=i_u+1}^{j_v} \tau_i}, \tau_{j_v}\Bigr)^{\prime}\Bigr)\mid \tau_i \colon i=i_u+1,\ldots,j_v\Bigr)\end{align*}

because of the independence between

\begin{equation*}\Bigl\{\Bigl(X_{\sum_{i=i_u+1}^{j_1} \tau_i}, \tau_{j_1}\Bigr)^{\prime},\ldots,\Bigl(X_{\sum_{i=i_u+1}^{j_v} \tau_i}, \tau_{j_v}\Bigr)^{\prime}\Bigr\}\end{equation*}

and $\{ \tau_i \colon i=1,\ldots,i_u \}$ . Thus the summand (3.3) is equal to zero because

\begin{equation*}\mathbb{E}(\tilde{F}(Y_{i_1},\ldots,Y_{i_u})\mid \tau_i\colon i=1,\ldots,i_u)\end{equation*}

and

\begin{equation*}\mathbb{E}\Bigl(\tilde{G}\Bigl(\Bigl(X_{\sum_{i=i_u+1}^{j_1} \tau_i}, \tau_{j_1}\Bigr)^{\prime},\ldots,\Bigl(X_{\sum_{i=i_u+1}^{j_v} \tau_i}, \tau_{j_v}\Bigr)^{\prime}\Bigr)\mid \tau_i \colon i=i_u+1,\ldots,j_v\Bigr) \Bigr)\end{equation*}

are independent.

The summand (3.2) is less than or equal to

\begin{align*}& \!\!\!\!\!\!\!\!\!\!\!\!\! \!\int_{\mathcal{I}^{j_v}} \Bigl| \mathrm{Cov}\Bigl(\tilde{F}\Bigl(\Bigl(X_{\sum_{i=1}^{i_1} s_i}, s_{i_1}\Bigr)^{\prime},\ldots,\Bigl(X_{\sum_{i=1}^{i_u} s_i}, s_{i_u}\Bigr)^{\prime}\Bigr), \\[5pt] &\tilde{G}\Bigl(\Bigl(X_{\sum_{i=1}^{j_1} s_i}, s_{j_1}\Bigr)^{\prime},\ldots,\Bigl(X_{\sum_{i=1}^{j_v} s_i}, s_{j_v}\Bigr)^{\prime}\Bigr) \Bigr) \Bigr| \, {\mathrm{d}} \mathbb{P}_{\{\tau_i\colon i=1,\ldots,j_v\}}(s_1,\ldots,s_{j_v}),\end{align*}

where $\mathbb{P}_{\{\tau \}}$ indicates the joint distribution of the sequence $\tau$ . For a given $(s_{1},\ldots,s_{j_v}) \in \mathcal{I}^{j_v}$ , we have $\tilde{F}( (\cdot, s_{i_1}),\ldots,(\cdot,s_{i_u})) \in \mathcal{F}$ and $\tilde{G}((\cdot, s_{j_1}),\ldots,(\cdot, s_{j_v})) \in \mathcal{G}$ . Since X is a $\Psi$ -weakly dependent process, the above inequality is less than or equal to

\begin{align*}&\int_{\mathcal{I}^{j_1-i_u}} C \, \Psi(\|\tilde{F}((\cdot, s_{i_1}),\ldots,(\cdot,s_{i_u}))\|_{\infty},\|\tilde{G}((\cdot, s_{j_1}),\ldots,(\cdot, s_{j_v}))\|_{\infty},\mathrm{Lip}(\tilde{F}((\cdot, s_{i_1}),\ldots,(\cdot,s_{i_u}))), \\[5pt] &\quad \mathrm{Lip}(\tilde{G}((\cdot, s_{j_1}),\ldots,(\cdot, s_{j_v}))),u,v) \, \epsilon \Biggl(\Biggl\|\sum_{i=i_u+1}^{j_1} s_i \Biggr\|\Biggr) {\mathrm{d}} \mathbb{P}_{\{\tau_i\colon i=1,\ldots,j_v\}}(s_1,\ldots,s_{j_v}),\end{align*}

and, because the sequence $\{\tau_i\colon i=i_u+1,\ldots,j_1\}$ is independent of the sequence $\{\tau_i\colon i=1,\ldots,i_u,j_1 +1,\ldots,j_v\}$ , the integral above is less than or equal to

\begin{align*}& \int_{\mathcal{I}^{j_1-i_u}} C \, \Psi(\|\tilde{F}\|_{\infty},\|\tilde{G}\|_{\infty},\mathrm{Lip}(\tilde{F}),\mathrm{Lip}(\tilde{G}),u,v) \\[5pt] &\quad \epsilon \Biggl(\Biggl\|\sum_{i=i_u+1}^{j_1} s_i \Biggr\|\Biggr) {\mathrm{d}} \mathbb{P}_{\{\tau_i\colon i=i_u+1,\ldots,j_1\}}(s_{i_u+1},\ldots,s_{j_1}).\end{align*}

We see that $j_1-i_u \geq n$ , and without loss of generality the coefficients $\epsilon$ are non-increasing. Thus we can conclude that the integral above is less than or equal to

\begin{equation*}C \, \Psi(\|\tilde{F}\|_{\infty},\|\tilde{G}\|_{\infty},\mathrm{Lip}(\tilde{F}),\mathrm{Lip}(\tilde{G}),u,v) \int_{\mathcal{I}} \epsilon(\|r\|) \mu^{*n}({\mathrm{d}} r).\end{equation*}

Finally, concerning $\theta$ -lex weak dependence defined in Remark 2.2, we obtain the following result.

Corollary 3.2. Let X be a strictly stationary and $\theta$ -lex weakly dependent random field defined on $\mathbb{R}^m$ , and let T be a renewal sampling sequence independent of X with values in $\mathbb{R}^m$ . Then Y is a strictly stationary process, and there exists a sequence $\mathcal{E}$ such that

(3.4) \begin{equation}|\mathrm{Cov}(\tilde{F}(Y_{i_1},\ldots,Y_{i_u}),\tilde{G}(Y_{j}))| \leq C \, \|\tilde{F}\|_{\infty}\;\mathrm{Lip}(\tilde{G}) \, \mathcal{E}(n)\end{equation}

for all

\begin{equation*}\begin{cases}(u,v) \in \mathbb{N}^* \times \mathbb{N}^*,\\[5pt] n\in \mathbb{N}, \\[5pt] \textit{$\{i_1,\ldots,i_u\} \subseteq \mathbb{Z}$ and $ j \in \mathbb{Z}$,} \\[5pt] \textit{with $ i_1\leq \ldots \leq i_u < i_u+n\leq j$,} \\[5pt] \textit{functions $ \tilde{F} \colon (\mathbb{R}^{d+1})^u \to \mathbb{R} $ and $ \tilde{G}\colon \mathbb{R}^{d+1} \to \mathbb{R} $ belonging to $\mathcal{\tilde{F}}$ and $\mathcal{\tilde{G}}$,}\end{cases}\end{equation*}

where C is a constant independent of n, and $\mathcal{E}$ is defined in (3.1).

Note that if the inequality (3.4) holds and the coefficients (3.1) are finite and converge to zero as n goes to infinity, then Y is a $\theta$ -weakly dependent process as defined in [Reference Dedecker and Doukhan19].

4. Explicit bounds for $\Psi$ -coefficients

In this section we consider renewal sampling of $X=(X_t)_{t \in \mathbb{R}}$ . Therefore the inter-arrival times $\tau$ are a sequence of non-negative i.i.d. random variables with values in $\mathbb{R}$ .

We first show that if X is $\Psi$ -weakly dependent and admits exponential or power decaying coefficients $\epsilon$ , then Y is, in turn, $\Psi$ -weakly dependent and its coefficients $\mathcal{E}$ preserve (at least asymptotically) the decay behavior of $\epsilon$ . This result directly enables the application of the limit theory for a vast class of $\Psi$ -weakly dependent processes Y, of which we present several examples throughout the section.

In fact central limit theorems for a $\Psi$ -weakly dependent process X typically hold under sufficient conditions of the following type: $\mathbb{E}[\|X_0\|^{\delta}]< \infty$ for some $\delta >0$ and the coefficients $\epsilon$ satisfy a condition of the form

(4.1) \begin{equation}\sum_{i=1}^\infty \epsilon(n)^{A(\delta)} < \infty,\end{equation}

where $A(\delta)$ is a certain function of $\delta$ . If X admits coefficients $\epsilon$ with exponential or sufficiently fast power decay, then conditions of type (4.1) are satisfied. If, in turn, Y is $\Psi$ -weakly dependent with coefficients having exponential or sufficiently fast power decay, then conditions of type (4.1) are also satisfied under renewal sampling.

4.1. Exponential decay

In terms of the Laplace transform of the inter-arrival times, we can obtain a general bound for the coefficients $(\mathcal{E}(n))_{n \in \mathbb{N}}$ .

Proposition 4.1. Let $X=(X_t)_{t \in \mathbb{R}}$ , $Y=(Y_i)_{i \in \mathbb{Z}}$ and $(T_i)_{i \in \mathbb{Z}}$ be as in Theorem 3.1. Let us assume that $\epsilon(r)\leq C\mathrm{e}^{-\gamma r}$ for $\gamma>0$ and denote the Laplace transform of the distribution function $\mu$ by

\begin{equation*}L_{\mu}(t)=\int_{\mathbb{R}^{+}} \mathrm{e}^{-tr} \mu({\mathrm{d}} r), \quad t \in \mathbb{R}_{+}.\end{equation*}

Then the process Y admits coefficients

\begin{equation*}\mathcal{E}(n)\leq C\biggl(\dfrac{1}{L_{\mu}(\gamma)}\biggr)^{-n},\end{equation*}

which converge to zero as n goes to infinity.

Proof of Proposition 4.1. We notice that $L_{\mu}(t) <1$ for $t>0$ and that $L_{\mu^{*n}}(t)=(L_{\mu}(t))^n$ ; see [Reference Sato41, Proposition 2.6].

Using the result obtained in Theorem 3.1, we have that

\begin{align*}\mathcal{E}(n)&=\int_{\mathbb{R}_+} \epsilon(r) \, \mu^{*n}({\mathrm{d}} r) \leq C \int_{\mathbb{R}_+} \mathrm{e}^{-\gamma r} \, \mu^{*n}({\mathrm{d}} r) = C L_{\mu^{*n}}(\gamma)=C (L_{\mu}(\gamma))^n.\end{align*}

As a direct consequence, if X is $\Psi$ -weakly dependent and admits exponentially decaying coefficients, the assumptions of Corollary 3.1 hold and Y inherits the asymptotic dependence structure of X under renewal sampling.

Example 4.1. If we have a renewal sampling with $\Gamma(\alpha,\beta)$ -distributed inter-arrival times for $\alpha, \beta >0$ , then $\mu^{*n}$ is the distribution function of a $\Gamma(n\alpha, \beta)$ -distributed random variable. By Proposition 4.1,

\begin{equation*}\mathcal{E}(n)=\int_{\mathbb{R}_+} \epsilon(r) \, \mu^{*n}({\mathrm{d}} r) \leq C \int_{(0,+\infty)} \mathrm{e}^{-\gamma r} \dfrac{\beta^{n\alpha}}{\Gamma(n\alpha)} r^{n\alpha-1} \mathrm{e}^{-\beta r} \, {\mathrm{d}} r = C \biggl( \dfrac{\gamma+\beta}{\beta} \biggr)^{-n\alpha}.\end{equation*}

A special case of the coefficients above is obtained for Poisson sampling, i.e. $\mu=\operatorname{Exp}\!(\lambda)$ with $\lambda>0$ . In this case, $\mu^{*n}$ is the distribution function of a $\Gamma(n,\lambda)$ -distributed random variable, and

\begin{equation*}\mathcal{E}(n)=\int_{\mathbb{R}_+} \epsilon(r) \, \mu^{*n}({\mathrm{d}} r) \leq C \int_{(0,+\infty)} \mathrm{e}^{-\gamma r} \dfrac{\lambda^n}{\Gamma(n)} r^{n-1} \mathrm{e}^{-\lambda r} \, {\mathrm{d}} r = C \biggl( \dfrac{\lambda+\gamma}{\lambda} \biggr)^{-n}.\end{equation*}

Remark 4.1. If the process X is $\Psi$ -weakly dependent with exponentially decaying coefficients, then the equidistant sampled process $(X_i)_{i \in \mathbb{Z}}$ has exponentially decaying coefficients $\epsilon(n)\leq C \mathrm{e}^{-\gamma n}$ for $\gamma >0$ and $n \in \mathbb{N}$ . By using the results in Example 4.1, we can design renewal sampling schemes such that the process Y has coefficients $\mathcal{E}$ with faster decay rate than the sequence of coefficients $\epsilon$ .

• For $\Gamma(\alpha,\beta)$ -distributed inter-arrival times, we obtain that the process Y has faster-decaying coefficients than $(X_i)_{i \in \mathbb{Z}}$ if the parameters $\alpha, \beta >0$ are chosen such that

  \begin{equation*}\biggl( \dfrac{\gamma+\beta}{\beta} \biggr)^{\alpha} \geq \mathrm{e}^{\gamma}.\end{equation*}

• In the case of Poisson sampling, the process Y admits faster-decaying coefficients than $(X_i)_{i \in \mathbb{Z}}$ if

  \begin{equation*}\lambda \geq \dfrac{\gamma}{\mathrm{e}^{\gamma}-1}.\end{equation*}
  The fraction appearing on the right-hand side of the inequality is less than 1 for all $\gamma >0$ . Therefore, because the average length of two adjacent observations is ruled by $\mathbb{E}[\tau_1]={1}/{\lambda}$ , we can design an (on average) lower sampling frequency scheme such that the coefficients $\mathcal{E}$ decay faster than $\epsilon$ , by choosing ${\gamma}/{(\mathrm{e^{\gamma}}-1)} \leq \lambda < 1$ .

4.2. Power decay

We now assume that the underlying process X is $\Psi$ -weakly dependent with coefficients $\epsilon(r)\leq C r^{-\gamma}$ for $\gamma>0$ .

We start with some concrete examples of inter-arrival time distributions $\mu$ (and therefore of renewal sampling sequences T), preserving the power decay of the coefficients $\epsilon$ .

Example 4.2. Let us consider renewal sampling with $\Gamma(\alpha,\beta)$ -distributed inter-arrival times for $\alpha, \beta >0$ . Then $\mu^{*n}$ is a $\Gamma(n\alpha, \beta)$ distribution. Thus

(4.2) \begin{equation}\mathcal{E}(n)=\int_{\mathbb{R}_+} \epsilon(r) \, \mu^{*n}({\mathrm{d}} r) \leq C \int_{(0,+\infty)} r^{-\gamma} \dfrac{\beta^{n\alpha}}{\Gamma(n\alpha)} r^{n\alpha-1} \mathrm{e}^{-\beta r} \, {\mathrm{d}} r = C \beta^{\gamma} \dfrac{\Gamma(n\alpha-\gamma)}{\Gamma(n\alpha)}.\end{equation}

For $n \to \infty$ , and applying Stirling’s series (see [Reference Tricomi and Erdélyi42]), we obtain that (4.2) is equal to $C \beta^{\gamma} n^{-\gamma} + {\mathrm{O}} (n^{-\gamma-1}).$

In the particular case of Poisson sampling, $\mu^{*n}$ is a $\Gamma(n,\lambda)$ distribution and

\begin{align*} \mathcal{E}(n)&=\int_{\mathbb{R}_+} \epsilon(r) \, \mu^{*n}({\mathrm{d}} r)\\[5pt] &\leq C \int_{(0,+\infty)} r^{-\gamma } \dfrac{\lambda^n}{\Gamma(n)} r^{n-1} \mathrm{e}^{-\lambda r} \, {\mathrm{d}} r \\[5pt] &= C \lambda^{\gamma} \dfrac{\Gamma(n-\gamma)}{\Gamma(n)}\\[5pt] & =C \lambda^{\gamma} n^{-\gamma}(1+{\mathrm{O}} (n^{-1})) \\[5pt] &= C \lambda^{\gamma} n^{-\gamma} + {\mathrm{O}} (n^{-\gamma-1}),\end{align*}

where the last equality holds as $n \to \infty$ .

Example 4.3. We let $\mathrm{Levy}(0,c)$ denote a Lévy distribution (see [Reference Zolotarev46, page 28]) with location parameter 0 and scale parameter c (a completely skewed $\frac{1}{2}$ -stable distribution). This distribution has infinite mean and variance. For $\mathrm{Levy}(0,c)$ -distributed inter-arrival times, we find that $\mu^{*n}$ is $\mathrm{Levy}(0,cn)$ . Thus

\begin{align*} \mathcal{E}(n)&=\int_{\mathbb{R}_+} \epsilon(r) \, \mu^{*n}({\mathrm{d}} r)\\[5pt] &\leq C\int_{\mathbb{R}_+} r^{-\gamma} \, \dfrac{({cn}/{2})^{{1}/{2}}}{\Gamma({1}/{2})} r^{-{3}/{2}} \mathrm{e}^{-{cn}/{2r}} \, {\mathrm{d}} r \\[5pt] &= C \dfrac{\Gamma({1}/{2}+\gamma)}{({cn}/{2})^{\gamma}\Gamma\bigl({1}/{2}\bigr)}\\[5pt] &= C \dfrac{\Gamma({1}/{2}+\gamma)}{\Gamma({1}/{2})} \biggl(\dfrac{c}{2}\biggr)^{-\gamma} n^{-\gamma}.\end{align*}

Example 4.4. We now consider the case where $\mu$ is an inverse Gaussian distribution with mean m and shape parameter $\lambda$ ( $IG(m,\lambda)$ for short). We see that $\mu^{*n}$ is an $IG(nm,n^2\lambda)$ distribution and

(4.3) \begin{align} \mathcal{E}(n)&= \int_{\mathbb{R}_+} \epsilon(r) \, \mu^{*n}({\mathrm{d}} r) \notag \\[5pt] &\leq C\int_{(0,+\infty)} r^{-\gamma} \biggl( \dfrac{n^2 \lambda}{2\pi r^3} \biggr)^{{1}/{2}}\exp\biggl(-\frac{n^2\lambda(r-nm)^2}{2n^2m^2r}\biggr) \, {\mathrm{d}} r \nonumber \\[5pt] &= n C \biggl( \dfrac{\lambda}{2\pi} \biggr)^{{1}/{2}} \exp\biggl(\frac{\lambda n}{m}\biggr) \int_{(0,+\infty)} r^{-\gamma-{3}/{2}}\exp\biggl(-\frac{\lambda n}{2 m}\biggl( \frac{r}{nm}+\frac{nm}{r}\biggr)\biggr) \, {\mathrm{d}} r \nonumber \\[5pt] &= C \biggl( \dfrac{\lambda}{2\pi} \biggr)^{{1}/{2}} m^{-\gamma-{1}/{2}} \, n^{-\gamma+{1}/{2}} \,\exp\biggl(\frac{\lambda n}{m}\biggr) \, 2 \, \mathcal{K}_{-\gamma-{1}/{2}}\biggl(\dfrac{\lambda n}{m}\biggr) \end{align}

after applying the substitution $x \;:\!=\; {r}/{(nm)}$ , and where $\mathcal{K}_{-\gamma-{1}/{2}}$ denotes a modified Bessel function of the third kind with order $-\gamma-\frac{1}{2}$ . Using the asymptotic expansion for modified Bessel functions from [Reference Jørgensen31, page 171], we obtain

\begin{equation*}\mathcal{K}_v(x)= \biggl(\dfrac{\pi}{2}\biggr)^{{1}/{2}} x^{-{1}/{2}} \mathrm{e}^{-x}(1+{\mathrm{O}} (x^{-1})).\end{equation*}

Thus, for $n \to \infty$ , (4.3) is equal to $({C}/{2}) m^{-\gamma}\, n^{-\gamma}+ {\mathrm{O}} (n^{-\gamma-1}).$

Example 4.5. Let the inter-arrival times follow a Bernoulli distribution with parameter $0 \leq p \leq 1$ . Then $\mu^{*n}$ is a $\operatorname{Bin}\!(n, p)$ distribution. If X admits coefficients $\epsilon(r)=C(1 \wedge r^{-\gamma})$ for $\gamma>0$ , then $\mathcal{E}(n)$ satisfies

(4.4) \begin{equation}\mathcal{E}(n)= \int_{\mathbb{R}_+} \epsilon(r) \, \mu^{*n}({\mathrm{d}} r) = C \biggl( (1-p)^n + \sum_{j=1}^n j^{-\gamma} \binom{n}{j} p^j (1-p)^{n-j} \biggr) . \end{equation}

For $n \to \infty$ , applying the asymptotic expansion proved in [Reference Wuyungaowa and Wang45, Theorem 1], we have that (4.4) is equal to $C (np)^{-\gamma} + {\mathrm{O}} (n^{-\gamma-1})$ .

Example 4.6. Let us consider inter-arrival times such that $\mu([0,k))=0$ for a fixed $k >0$ . Then, straightforwardly,

\begin{equation*}\mathcal{E}(n)=\int_{\mathbb{R}_+} \epsilon(r) \, \mu^{*n}({\mathrm{d}} r) \leq C (nk)^{-\gamma}.\end{equation*}

In Examples 4.2, 4.4, and 4.5 we obtain asymptotic bounds for the coefficients $\mathcal{E}$ , whereas we have exact ones in Examples 4.3 and 4.6. For a general inter-arrival time distribution we can just show that the coefficients $\mathcal{E}$ decay at least (asymptotically) with the same power. This result relies on the following lemma.

Lemma 4.1. Let $\mu, \nu$ be two probability measures on $\mathbb{R}^{+}$ such that $\mu([0,b)) \leq \nu([0,b))$ for all $b >0$ and let $f\colon \mathbb{R}^{+} \to \mathbb{R}^{+}$ be non-increasing. Then

\begin{equation*}\int_{\mathbb{R}_+} f(r) \mu^{*n}({\mathrm{d}} r) \leq \int_{\mathbb{R}_+} f(r) \, \nu^{*n}({\mathrm{d}} r).\end{equation*}

Proof of Lemma 4.1. The proof follows by applying measure-theoretic induction.

Proposition 4.2. Let $X=(X_t)_{t \in \mathbb{R}}$ , $Y=(Y_i)_{i \in \mathbb{Z}}$ , and $(T_i)_{i \in \mathbb{Z}}$ be as in Theorem 3.1. Let us assume that $\epsilon(r)\leq C r^{-\gamma}$ for $\gamma>0$ . Let $a >0$ be a point in the support of $\mu$ such that $\mu([0,a))>0$ , and set $p=\mu([a,\infty])$ . Then the process Y admits coefficients $\mathcal{E}(n)\leq C (n a p)^{-\gamma}$ as $n \to \infty$ .

Proof of Proposition 4.2. Let us assume without loss of generality that $\mu\neq \delta_a$ (otherwise Example 4.6 applies for any $a \in \mathbb{R}_{+}$ ), where $\delta_a$ denotes the Dirac delta measure for $a \in \mathbb{R}_+$ . Set $\nu= p \delta_{a}+(1-p) \delta_0$ . The latter is a Bernoulli distribution that assigns probability p to the inter-arrival time a and $(1-p)$ to the time 0. It follows that $\mu([0,b))\leq \nu([0,b))$ for all $b >0$ . Then, by using Lemma 4.1, the result in Example 4.5, and [Reference Wuyungaowa and Wang45, Theorem 1],

\begin{align*} \mathcal{E}(n) &\leq C \int_{\mathbb{R}_+} r^{-\gamma} \mu^{*n}({\mathrm{d}} r) \\[5pt] &\leq C \int_{\mathbb{R}_+} r^{-\gamma} \nu^{*n}({\mathrm{d}} r) \\[5pt] &= C \Biggl( (1-p)^n + \sum_{j=1}^n (a j)^{-\gamma} \binom{n}{j} p^j (1-p)^{n-j} \Biggr) \\[5pt] &= C (n a p)^{-\gamma} + {\mathrm{O}} (n^{-\gamma- 1}),\end{align*}

where the last inequality holds for $n \to \infty$ .

Remark 4.2. Proposition 4.2 gives us an upper bound for the coefficients $\mathcal{E}$ . This means that the true decay of the coefficients $\mathcal{E}$ could be faster, in general, than $n^{-\gamma}$ . However, we have not found examples of sequences $\tau$ where this happens. Even for extremely heavy-tailed inter-arrival time distributions such as in Example 4.3, we can just find an estimate from above of the coefficients of the renewal sampled process Y, i.e. $\mathcal{E}(n) \leq C n^{-\gamma}$ for large n, that has the same power decay as the coefficients $\epsilon$ .

Proposition 4.2 summarizes the results given in this section. In fact, as long as X is $\Psi$ -weakly dependent such that there exists a $\gamma >0$ with $\epsilon(r) \leq C r^{-\gamma}$ , then the assumptions of Corollary 3.1 are satisfied and Y inherits the asymptotic dependence structure of X. Note that Proposition 4.2 ensures that Y is $\Psi$ -weakly dependent also when, for example, $\epsilon(r)=C{(r \log\!(r))}^{-1}$ and then $\epsilon(r) \leq C n^{-1}$ . Therefore caution has to be exercised when checking conditions of type (4.1) for the process Y.

Example 4.7. Let us consider the sufficient condition for the applicability of the central limit theorem for $\kappa$ -weakly dependent processes (see [Reference Doukhan and Wintenberger26]), where (4.1) holds with $A(\delta)=1$ . If X is a $\Psi$ -weakly dependent process with coefficients $\epsilon(r)=C{(r \log\!(r))}^{-1}$ , then Y is a $\Psi$ -weakly dependent process with coefficients $\mathcal{E}(n) \leq \tilde{C} n^{-1}$ as $n \to \infty$ by applying Proposition 4.2. We have that the coefficients $\epsilon(r)$ are summable and satisfy (4.1), but we do not know the summability of the coefficients $\mathcal{E}(n)$ , as Proposition 4.2 just gives an upper bound of their value, which is not summable.