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

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.

On natural convergence rate estimates in the Lindeberg theorem

“…Hence, it would be useful to have a possibility to balance the contribution of the terms |M n | and L n to optimize the resulting bound. Similarly to (10) and (11), let us introduce a balancing parameter γ > 0 and for ε > 0 and g ∈ G denote…”

“…Observe that, in the i.i.d. case, the fractions L e,n (1, 1) and L e,n (∞, 1) coincide with the Esseen fractions in ( 6) and ( 7), respectively, and L r,n (1, 1) coincides with the Rozovskii fraction in (8), so that (11). Recall that γ * = 0.5599 .…”

“…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…”

“…It is easy to see that ( 12) and ( 13) with g(z) = g * (z)(≡ z) reduce, respectively, to (10) and (11) with γ = 1 and…”

“…2.73, C r (1) = A r (1, 1) 2.73, inequalities (10), (11), (12), and (13) may be much sharper than (2) and (4) due to the more favorable dependence of the appearing fractions on truncated third order moments |M n ( • )|, which vanishes, say, in the symmetric case, or for even n and oscillating sequence X k d = (−1) k X, k = 1, . .…”