On Convergence Rate in the Local Limit Theorem for Densities Under Various Moment Conditions

Abstract

We refine certain estimates of convergence rate in the local central limit theorem for the densities of sums of independent identically distributed random variables possessing finite absolute moments up to the order 2 + δ, where δ is some number from the half-interval (0, 1]. Along with the uniform estimates we obtain non-uniform estimates of the first, second, and third order (for δ = 1), and the estimates in the L_p metrics. The obtained estimates have the form of the sum of two terms, the first of which is the Lyapunov fraction of the corresponding order with the coefficient depending only on δ, and the second one exponentially decays with the growth of the number of summands. The values of the coefficient in the Lyapunov fraction are considerably smaller than the known ones.