Poisson type approximations for the Markov binomial distribution

Abstract

The Markov binomial distribution is approximated by the Poisson distribution with the same mean, by a translated Poisson distribution and by two-parametric Poisson type signed measures. Using an adaptation of Le Cam's operator technique, estimates of accuracy are proved for the total variation, local and Wasserstein norms. In a special case, asymptotically sharp constants are obtained. For some auxiliary results, we used Stein's method.

Binomial Approximation to the Markov Binomial Distribution

“…214-215] and are already proved inČekanavičius and Roos [6]. For the proof of (15) note that (13), (14) and the trivial fact U = 2 imply that…”

“…Applying (19) and Lemma 3.5, we get Combining the last two estimates, we get (6). The local estimate is proved using (10).…”

“…seeČekanavičius and Roos [6]. If p and q are uniformly bounded away from unity, then (p − q) n is at least of exponentially vanishing order.…”

Asymptotics for the sum of three state Markov dependent random variables

“…The compound Poisson approximation is frequently used to approximate aggregate losses in risk models (see, for example, [5,8,9,12,14,21]); however, in those models it is usually assumed that rvs are independent of time period n ∈ N. The compound Poisson approximation to sums of Markov dependent rvs was investigated in [6]. Numerous papers were devoted to Markov Binomial distribution, see [1,3,4,7,10,18,19], and the references therein. It seems, however, that the case of Markov chain containing absorbing state was not considered so far.…”

Multimodality of the Markov Binomial Distribution

“…Obtaining var ν (K n ) is more involved, because of correlations. A closed formula for the variance in the following proposition has been obtained by several authors (see, for example [2] and [13]). Here it is particularly compact by our use of excentricities.…”