Compound Poisson approximations for sums of 1-dependent random variables. I

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

“…Next steps in the proof are very similar to the steps from the proof of Lemma 7 in [18] and, therefore, will not be discussed in detail. By induction, we prove that |f k (u) − 1| Cp 2 |e u − 1| 2 , then we prove |f k (u) − 1 − p 2 (e u − 1)| Cp 3 |e u − 1| 3 , and finally obtain…”

On magic factors in Stein’s method for compound Poisson approximation

“…As noted by [2], if q and λ are small, (3) will suffice for providing a bound on M (K) 1 . In the case of larger λ, [2] considers the use of the bound in (8), noting that θ 0 = (n−k +1) 2 ψ and θ 1 ≤ 4q k θ 0 , so that (7) is satisfied if q k < 1/8. Under this condition, we use (8) to obtain M…”

“…. , ξ n be independent Bernoulli random variables, each with mean p. Let W = n i=1 ξ i ξ i+1 count the number of 2-runs in this sequence, where all indices are treated modulo n. Compound Poisson approximation for W is a well-studied problem (see, for example, [2,4,7,8] and references therein), so gives us an excellent application within which to examine the benefit of our Theorem 2.…”

Approximation Methods in Probability Theory

“…Heinrich introduced his method in [69,70] and extended it to Markov chains [71] and random vectors [72], see also an overview in [73]. 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].…”

“…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].…”