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Removing logarithms in Poisson approximation to the partial sum process of a Markov chain. (English) Zbl 1295.60062

The paper considers a Markov chain \(\{X_k\}_{k\geq0}\) with finite state space \(\{0, 1, \dots, m\}\), irreducible transition matrix \(P=(p_{ij})\), stationary distribution \(\pi=(\pi_0, \pi_1,\dots, \pi_m)\) and initial condition \(X_0=0\). For each \(n\geq 1\), denote the \(n\)-step transition matrix of \(P\) by \(P^{(n)}= (p^{(n)}_{ij})\). For each \(n\geq 0\) and \(0\leq t\leq 1\), define the partial sum process \(Z_n(t) = \sum_{k=0}^{[nt]}X_k\, \delta_{k/n}\), where \([n t]\) denotes the integer part of \(n t\). It is known that if \(n\,(1-p_{00}^{(n)})\to \lambda\), \(\max_{1\leq j\leq m}(1-p_{j0}^{(n)}) \to 0\), \({{p_{01}^{(n)}}\over{1-p_{00}^{(n)}}}\to 1\), then \(Z_n\) converges in law to a homogeneous Poisson process with rate \(\lambda\), see [S. He and J. Wang [in: Dirichlet forms and stochastic processes. Proceedings of the international conference held in Beijing, China, 1993. Berlin: de Gruyter. 175–184 (1995; Zbl 0839.60063)]. S. He and A. Xia [Stochastic Processes Appl. 68, No. 1, 101–111 (1997; Zbl 0889.60049)], using the maximal coupling and the Wasserstein process of Markov chains, gave some error bounds for a Wasserstein distance between the distributions of the partial sum process and a Poisson process on the positive half-line. They evaluated how well the partial sum process can be approximated in distribution by a Poisson process in terms of the transition probabilities \((p_{ij})\) and the stationary distribution \(\pi\). However, all those error bounds increase logarithmic with the mean of the Poisson process. Certainly, removing logarithms from those error bounds is of both theoretical and practical interest. In this respect, [T. C. Brown et al., Stochastic Processes Appl. 87, No. 1, 149–165 (2000; Zbl 1045.60045)] gave a bound controlled by Palm processes and applied it to a simple Bernoulli process, a randomly shifted Bernoulli process and networks of queues. However, the argument of [Zbl 1045.60045] does not work in the present situation of the partial sum process of a Markov chain. As a matter of fact, the partial sum process is much more complicated than those models considered in [Zbl 1045.60045]. Nevertheless, using the coupling method as in [Zbl 0889.60049] and a general result for estimating the errors of Poisson process approximation in [T. C. Brown and A. Xia, Ann. Probab. 29, No. 3, 1373–1403 (2001; Zbl 1019.60019)], the author of the present paper gives a new error bound for the Wasserstein distance between the distributions of the partial sum process and a Poisson process. Indeed, the new error bound has no logarithm anymore and it is bounded and asymptotically remains constant as the mean increases.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60E15 Inequalities; stochastic orderings
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