A Generalization of the Rozovskii Inequality

On natural convergence rate estimates in the Lindeberg theorem

“…with C e (ε) A e (min{1, ε}, 1), C e (+0) = ∞, so that A 6 C e (∞) with the equality in the i.i.d. case, and inequality (11) with γ = 1 was generalized in [9] to…”

“…and have finite third-order moments. Then, due to(22), we havelim ε→∞ L r,n (g 0 , ε, γ) = lim ε→∞ γ ε |M n (ε)| + sup 0<z<1 zL n (z) = sup 0<z<1 zL n (z).On the other hand, in[9, Theorem 3] it is shown that sup…”

Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem

“…The most common approximation is the normal one which is based on the central limit theorem. 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 (in terms of full moments and under the additional moment-type assumptions), or Osipov-Petrov's [3,4], Esseen's [5], Rozovskii's [6], Wang-Ahmad's [7] inequalities and their generalizations [8][9][10][11] (in terms of truncated moments without any additional assumptions). However, the most natural estimates, such as Esseen's, Rozovskii's and Wang-Ahmad's inequalities contained unknown constants, and their application in practice was made possible only by the results of [8,11], where in particular, the unknown constants in the above inequalities were evaluated.…”

“…Let us denote by G a set of all non-decreasing functions g : [0, ∞) → [0, ∞) such that g(z) > 0 for z > 0 and z/g(z) is also non-decreasing for z > 0. The set G was initially introduced by Katz [15] and used later in the works [4,[9][10][11][12]16]. In [9] it was proved that (i) For every function g ∈ G and a > 0…”

Asymptotically exact constants in natural convergence rate estimates in the Lindeberg theorem

“…Let us denote by G a set of all non-decreasing functions g : [0, ∞) → [0, ∞) such that g(z) > 0 for z > 0 and z/g(z) is also non-decreasing for z > 0. The set G was initially introduced by Katz [15] and used later in the works [20,18,17,11,10,12]. In [11] it was proved that (i) For every function g ∈ G and a > 0…”