On the convergence of Markov binomial to Poisson distribution (Q1613105)

Let \(\{X_i,\;i\geq 0\}\) be a Markov chain on the two states 0 and 1, and \(S_n\) the number of visits to 1. A number of authors, starting with \textit{B. O. Koopman} [Proc. Natl. Acad. Sci. USA 36, 202-207 (1950; Zbl 0037.08502)], have considered Poisson approximation to the distribution of \(S_n\), when \(nP[X_1=1 \mid X_0=0] \to\lambda\) and \(P[X_1=1 \mid X_0=1]\to 0\). In this paper, the author strengthens the usual total variation approximations, by proving approximation with respect to a metric in which discrepancies in the upper tail receive exponentially increased weighting; cf. \textit{G. Simons} and \textit{N. L. Johnson} [Ann. Math. Stat. 42, 1735-1736 (1971; Zbl 0235.60033)].