Complementary asymptotic analysis for a minimal random walk

TL;DR

This paper provides a comprehensive asymptotic analysis of the minimal random walk, including an almost sure central limit theorem, quadratic strong laws, and alternative proofs of functional limit theorems using Pólya urn schemes.

Contribution

It introduces new asymptotic results for the minimal random walk and offers alternative proof methods based on Pólya urn schemes.

Findings

Almost sure central limit theorem established

Quadratic strong laws generalized

Functional limit theorems proved via Pólya urn approach

Abstract

We discuss a complementary asymptotic analysis of the so called minimal random walk. More precisely, we present a version of the almost sure central limit theorem as well as a generalization of the recently proposed quadratic strong laws. In addition, alternative demonstrations of the functional limit theorems will be supplied based on a P\'olya urn scheme instead of a martingale approach.