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Some notes on Lamperti's recurrence of stochastic sequences

Vyacheslav M. Abramov

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].

Some notes on Lamperti's recurrence of stochastic sequences

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].

Paper Structure

This paper contains 3 sections, 3 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Main result and its proof
  3. Discussion

Key Result

Theorem 1.1

(Lamperti L.) Let the nonnegative stochastic sequence $X_n$ satisfy 1 and 2, and, as $x\to\infty$, Then the stochastic sequence is recurrent. If instead for some $\theta>1$ and almost all large $x$, then the stochastic sequence is transient.

Theorems & Definitions (6)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Example 2.1
  • Theorem 2.2
  • proof