On large deviations for sums of discrete m-dependent random variables

Abstract

The ratio P (S n = x)/P (Z n = x) is investigated for three cases: (a) when S n is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and Z n is Poisson rv; (b) when S n is a statistic of 2-runs and Z n is negative binomial rv; and (c) when S n is statistic of N (1, 1)-events and Z n is a binomial r.v. We also consider the approximation of P (S n x) by Poisson distribution with parameter depending on x.

On moderate deviations in Poisson approximation

“…Hence it is of practical interest to consider their approximations via moderate deviations in Poisson approximation in a similar fashion to (1.2). However, there is not much progress in the general framework except the special cases in [16], [8], [18], [34], and [13]. This is partly due to the fact that the tail behaviour of a Poisson distribution is significantly different from that of a normal distribution, and this fact was observed by Gnedenko [25] in the context of extreme value theory.…”

On moderate deviations in Poisson approximation

“…Hence it is of practical interest to consider their approximations via moderate deviations in Poisson approximation as in the moderate deviation theorem (1.2). However, there is not much progress in the general framework except the special cases in [Chen & Choi (1992), Barbour, Chen & Choi (1995), Chen, Fang & Shao (2013a), Tan, Lu & Xia (2018), Čekanavičius & Vellaisamy (2019)]. The titles of [Barbour, Chen & Choi (1995), Chen, Fang & Shao (2013a)] without further progress indicate that the topic itself is generally too hard to make noteworthy contribution.…”