On Natural Convergence Rate Estimates in the Lindeberg Theorem

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

“…The main goal of the present work is construction of the lower bounds of the absolute constants C E (ε, γ), C R (ε, γ) in inequalities (11), (12), and also of the constants A E (ε, γ), A R (ε, γ) in inequalities (1), (2), in particular, we show that even in the i.i.d. case…”

“…Following [16,7,27,3,4,5,6,22,23,24], let us define exact, asymptotically exact and asymptotically best constants in inequalities ( 11), (12). Let F be a set of all d.f.…”

“…According to the terminology, introduced by Zolotarev [27], such convergence rate estimates are called natural. Together with (1), inequality (2) is surely a natural convergence rate estimate in the Lindeberg-Feller theorem under the additional assumption of existence of such an ε 0 > 0 that M n (ε 0 ) = 0 for all sufficiently large n (see, e.g., [12]), in particular, if the r.v. 's X 1 , .…”

“…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…”

“…A detailed survey and analysis of the relationships between inequalities (1), ( 2), ( 11), (12) with inequalities of Katz [15], Petrov [20], Osipov [19], Esseen [8], Rozovskii [21], and Wang-Ahmad [26] can be found in papers [9,12].…”