On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

TL;DR

This paper proves that under certain conditions, the normalized moments of vector martingales of even order converge almost surely, and applies these results to statistical models like autoregressive and branching processes.

Contribution

It establishes almost sure convergence of moments for vector martingales and demonstrates their application in statistical models, extending previous asymptotic results.

Findings

Normalized moments of even order converge almost surely.

New asymptotic properties for estimation and prediction errors.

Applications to autoregressive models and branching processes.

Abstract

We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure cental limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.