A Compound Poisson Convergence Theorem for Sums of $$m$$ m -Dependent Variables

Abstract

We prove the Simons-Johnson theorem for the sums S n of m-dependent random variables, with exponential weights and limiting compound Poisson distribution CP(s, λ). More precisely, we give sufficient conditions for ∞ k=0 e hk |P (S n = k)−CP(s, λ){k}| → 0 and provide an estimate on the rate of convergence. It is shown that the Simons-Johnson theorem holds for weighted Wasserstein norm as well. The results are then illustrated for N (n; k 1 , k 2 ) and k-runs statistics.

Two limit theorems for Markov binomial distribution

“…Similar results hold for convolutions on a measurable Abelian group and sums of m-dependent random variables; see [5] and [15].…”

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

“…and it remains to apply (11). The estimates (16) are proved in [7], Lemma 4.3. The estimates (18) and ( 19) follow from the expansion in factorial moments, namely,…”

“…. , K X j =    1, with probability mα(p), 0, with probability 1 − mα(p), ν 2 (j) = ν 2 (K + 1) = 0, ν 1 (j) = mα(p) and EX 1 X 2 = α(p) 2 m(m + 1)/2, see [7].…”

Approximation Methods in Probability Theory

“…Hwang in [77] applied the inversion formula for the moment generating function via a Cauchy integral and used asymptotic analysis of complex functions, proving estimates of the form o.1/. Lemma 5.1 was applied for sums of 1-dependent random variables in [45].…”

“…Heinrich's method for Poisson-type approximations was adapted in [103]. Further results for compound Poisson, binomial and negative binomial approximations of m-dependent integer-valued random variables can be found in [45]. In this section we used special cases of Theorem 1 from [104] and Theorem 2 from [103].…”