在中等偏差区域达到固定水平的随机漫步最大时刻的极限定理

IF 0.6 4区数学 Q3 MATHEMATICS Mathematical Notes Pub Date : 2024-07-05 DOI:10.1134/s0001434624030192

M. A. Anokhina

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引用次数: 0

摘要

摘要我们考虑一种具有零均值和有限方差的随机漫步，其步长为算术级数。关于行走达到最大值的时间的 arcsine 定律是众所周知的。在本文中，我们将在最大值本身固定的假设条件下考虑达到最大值时刻的分布。我们的研究表明，在最大值偏差适中的情况下，最大值时刻的分布经过适当的归一化后，会趋近于一个自由度的秩方分布。在非网格情况下也得到了类似的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Limit Theorem for the Moment of Maximum of a Random Walk Reaching a Fixed Level in the Region of Moderate Deviations

Abstract

We consider a random walk with zero mean and finite variance whose steps are arithmetic. The arcsine law for the time the walk reaches its maximum is well known. In this paper, we consider the distribution of the moment of reaching the maximum under the assumption that the maximum value itself is fixed. We show that, in the case of a moderate deviation of the maximum, the distribution of the moment of the maximum with appropriate normalization converges to the chi-square distribution with one degree of freedom. Similar results are obtained in the nonlattice case.

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来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.