Abstract
Following (Shevtsova, 2013) we introduce detailed classification of the asymptotically exact constants in natural estimates of the rate of convergence in the Lindeberg central limit theorem, namely in Esseen’s, Rozovskii’s, and Wang–Ahmad’s inequalities and their structural improvements obtained in our previous works. The above inequalities involve algebraic truncated third-order moments and the classical Lindeberg fraction and assume finiteness only the second-order moments of random summands. We present lower bounds for the introduced asymptotically exact constants as well as for the universal and for the most optimistic constants which turn to be not far from the upper ones.
Highlights
In various applications of probability theory, one has to approximate an unknown distribution of a sum of independent random variables with some known law
The adequacy of the normal approximation can be estimated with the help of convergence rate estimates in the central limit theorem such as the celebrated Berry–Esseen [1,2] inequality, or Osipov–Petrov’s [3,4], Esseen’s [5], Rozovskii’s [6], Wang-Ahmad’s [7] inequalities and their generalizations [8,9,10,11]
Since the crucial role in estimation of the adequacy of the normal approximation is played by the values of appearing absolute constants, it is very important to understand how accurate the existing upper bounds for the constants are, how much they might be lowered and if it is worth trying to improve the method of their evaluation
Summary
In various applications of probability theory, one has to approximate an unknown distribution of a sum of independent random variables with some known law. Let us note that estimate (2) with ε = γ = 1 coincides with the Rozovskii inequality [6] ([Corollary 1]) and establishes an upper bound for the appearing absolute constant. Estimate (1) yields upper bounds for the absolute constants in Esseen’s inequalities. In order not to introduce excess indexes, we use identical notation for the asymptotic constants in (11), (12), in what follows every time specifying the inequality in question. CAB ( g1 , ε, γ) and CAE ( g1 , ε, γ) are not defined in any of the inequalities (11), (12), since the corresponding fractions LE,n ( g1 , ε, γ), LR,n ( g1 , ε, γ) are bounded from below by one uniformly with respect to ε and γ (see (13) and (14)) and, cannot be infinitesimal. Where we omitted the arguments g, ε, γ for clarity)
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