Tail estimates for martingale under "LLN" norming sequence

TL;DR

This paper derives non-asymptotic exponential and moment tail estimates for discrete-time martingales normalized by 1/n, demonstrating their accuracy and applicability in the context of the Law of Large Numbers.

Contribution

It provides the first non-asymptotic tail estimates for martingales under LLN norming sequences, with proofs of their exactness.

Findings

Derived explicit exponential tail bounds for martingales

Established the accuracy of these tail estimates

Applied results to classical LLN normalization context

Abstract

In this paper non-asymptotic exponential and moment estimates are derived for tail of distribution for discrete time martingale under norming sequence 1/n, as in the classical Law of Large Numbers (LLN), by means of martingale differences as a rule in the terms of unconditional moments and tails of distributions of summands. We show also the exactness of obtained estimations.