Some notes on Lamperti's recurrence of stochastic sequences

TL;DR

This paper revisits Lamperti's theorem on the recurrence and transience of stochastic sequences, connecting it with recent criteria for Markov chain recurrence, offering new insights into stochastic process behavior.

Contribution

It establishes a novel connection between Lamperti's theorem and recent recurrence criteria for Markov chains with countable states.

Findings

Links Lamperti's theorem to new recurrence criteria

Provides a deeper understanding of stochastic sequence behavior

Enhances theoretical framework for Markov chain recurrence

Abstract

The present study provides another look on Lamperti's theorem on recurrence or transience of stochastic sequences. We establish connection between Lamperti's theorem and the recent result by the author [V. M. Abramov, A new criterion for recurrence of Markov chains with infinitely countable set of states. \emph{Theor. Probab. Math. Stat.} \textbf{112} (2025), 1--15].