Stretched random walks and the behaviour of their summands

2012/05/27 by Michel Broniatowski, Broniatowski, Michel, Zhansheng Cao +1 · 1 citation
Computer Science · Economics, Econometrics and Finance · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics #math.PR #math.ST #stat.TH

paper · pdf · doi:10.48550/arxiv.1205.5936

arxiv created 2012/05/27 · openalex publication_date 2012/05/27 · arxiv updated 2012/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper explores the joint behaviour of the summands of a random walk when their mean value goes to infinity as its length increases. It is proved that all the summands must share the same value, which extends previous results in the context of large exceedances of finite sums of i.i.d. random variables. Some consequences are drawn pertaining to the local behaviour of a random walk conditioned on a large deviation constraint on its end value. It is shown that the sample paths exhibit local oblic segments with increasing size and slope as the length of the random walk increases.

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