Almost Sure Invariance Principles via Martingale Approximation

TL;DR

This paper develops a method to approximate stationary processes with martingales, enabling improved proofs of classical limit theorems like the CLT and LIL, with broad applicability and easier verification.

Contribution

It introduces a new estimate for martingale approximation of stationary processes, leading to stronger almost sure invariance principles and improved limit theorem results.

Findings

Enhanced almost sure invariance principles for stationary processes

Broader applicability to linear processes and Bernoulli shifts

Improved conditions for CLT and LIL approximations

Abstract

In this paper we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale with stationary differences. The results are exploited to further investigate the central limit theorem and its invariance principle started at a point, as well as the law of the iterated logarithm via almost sure approximation with a Brownian motion, improving the results available in the literature. The conditions are well suited for a variety of examples; they are easy to verify, for instance, for linear processes and functions of Bernoulli shifts.