On the constants in the uniform and non-uniform versions of the Berry-Esseen inequality for Poisson random sums

Abstract

The sums of large number of independent random variables are very popular mathematical models for many real objects. The central limit theorem states that the distribution of such sum must approximately fit the normal distribution under a broad range of realistic conditions. The normal approximation is valid as long as the tails of the distribution are not too heavy, so that the variance were finite. Moreover, if the random summands have the moments of order higher than 2, then the normal approximation becomes more precise. The most interesting case is when the moment order lies between 2 and 3: the central limit theorem is still valid, but the random summands have so heavy tails that the third-order moment does not exist. Such heavytailed distributions are used, for example, for the analysis of the telecommunication system traffic. The present paper is devoted to the accuracy estimation of the normal approximation just in that case. We will present two-sided bounds for the constant in the Berry-Esseen inequality for Poisson random sums of independent identically distributed random variables with the finite moment order that lies between 2 and 3. The lower estimates obtained for the first time. We will improve the lower estimates and prove non-uniform estimates.

On the Accuracy of the Normal Approximation to Compound Poisson Distributions

“…Let us introduce also the upper asymptotically exact constant for symmetric distributions C AE (F 2+δ,s ) similarly to C AE (F 2+δ ) with F 2+δ replaced by F 2+δ,s . In the works [30], [10] lower bounds were found for C AE (F 2+δ ) with 0 < δ 1, but since the corresponding extremal distributions are symmetric, the same bounds remain valid for C AE (F 2+δ,s ) as well. Thus, from the results of [30], [10] (for 0 < δ 1) and Theorem 7 below (for δ = 0) it follows that for all 0 δ 1 2 and γ δ/2 ≡ 1 for δ = 0.…”

“…(in the formulations of the corresponding theorems in the works [30], [9] the lower bounds for C AE (F 2+δ ) were announced, but, in fact, the lower bounds for C AE (F 2+δ,s ) were obtained). The values of the minorant in (9) are given in Table 2 for some 0 < δ < 1.…”

“…Apparently, the upper bound for the constant C BE (0) can be sharpened, since the upper bound for sup x |F (x) − Φ(x)| used in the present work for this purpose is attained at a two-point distribution F ∈ F 2 which is not infinitely divisible and thus cannot be compound Poisson. Notice that the lower bounds for C BE (δ), 0 < δ 1, established in [30], [10] and in Theorem 5 below in terms of the upper asymptotically exact constants C AE (F 2+δ ), C AE (F 2+δ,s ), generally speaking, do not remain valid if we restrict ourselves to the consideration of the class of d.f. 's F ∈ F 2+δ in (1) with small or infinitely small values of the Lyapunov fraction λ (F ).…”