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Limit Properties of Record Numbers in Random walks

Penghui Lu, Yuqiang Li, Qiang Yao

Abstract

In this paper, we systematically summarize and enhance the understanding of weak convergence and functional limits of record numbers in discrete-time random walks under Spitzer's condition, and extend these findings to $σ$--record numbers using similar methods. Additionally, we identify a sufficient condition for the existence of functional limits for record numbers in continuous-time random walks. Finally, we derive corresponding results for large deviations, moderate deviations, and laws of the iterated logarithm pertaining to record numbers in discrete-time random walks.

Limit Properties of Record Numbers in Random walks

Abstract

In this paper, we systematically summarize and enhance the understanding of weak convergence and functional limits of record numbers in discrete-time random walks under Spitzer's condition, and extend these findings to --record numbers using similar methods. Additionally, we identify a sufficient condition for the existence of functional limits for record numbers in continuous-time random walks. Finally, we derive corresponding results for large deviations, moderate deviations, and laws of the iterated logarithm pertaining to record numbers in discrete-time random walks.
Paper Structure (5 sections, 9 theorems, 64 equations)

This paper contains 5 sections, 9 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Weak convergence of $R_n$
  3. Large, Moderate Deviations and Laws of the Iterated Logarithm
  4. Proof of Weak Convergence and Functional Limits
  5. Proof of Large, Moderate Deviation Principles and Laws of the Iterated Logarithm

Key Result

Theorem 1

(Weak convergence and functional limits of record numbers in discrete-time random walks) Write $g_{\rho}$ as the Mittag-Leffler distribution with parameter $\rho$, $0\leq \rho \leq1$. There exists some $u(n)>0$ such that $R_n/u(n)$ has a weak convergence limit if and only if $\mathrm{P}\left(S_n\ge0 Additionally, Bingham notes that the process $R$ would also have a functional limit, which is the i

Theorems & Definitions (12)

  • Theorem 1
  • Example 1
  • Example 2: An example in Paulauskas bib:stable law
  • Example 3: Li and Yao bib:eleven
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Theorem 6: Large deviations
  • Theorem 7: Moderate deviations
  • ...and 2 more