Asymptotic expansions for the probabilities of large deviations. Normal approximation. III

On a centered Poisson approximation

“…As proyed by Alelkevieien6 and Statulevi~ius [7], analogues of condition (S) can also be formulated for other infinitely divisible approximations. Note that in 1976, Bikelis and 7_,emaitis [11] formulated the following analogue of (S):…”

“…Large deviations and integral estimates under condition (P) were considered in [12], [22], [23], [29], [30] and, under (S), in [3]- [5]. (Note that, in [5], Ale~kevi~ien6 and Statulevi~ius obtained results under a considerably weaker form of (S), where k!…”

“…Large deviations under conditions (S) or (P) were considered in [3], [5], [12], [16], [30]. In all these papers the approximating distribution (or measure) depended on the cumulants of F only.…”

“…In recent years, limit theorems have been marked by numerous papers on Poisson approximations; see, for example, [1]- [5], [7], [9], [11], [13], [15], [17]- [24], [29]- [33] and the references therein. We can state that, in this field, an overwhelming majority of research is following traditions established by the pioneering papers of Prokhorov [32], Le Cam [26], Grigelionis [20], and Franken [19].…”

Large deviations for integer centered poisson approximation

“…The Poisson distribution is one of the most popular approximations (see, [1]- [7], [9]- [10], [12]- [21] and references therein). As a rule, it is applied only to the distributions of rare events, i.e., under conditions which are close to the conditions sufficient for the existence of the Poisson limiting law.…”