A conditional limit theorem for random walks under extreme deviation

2012/06/29 by Michel Broniatowski, Broniatowski, Michel, Zhansheng Cao +1 · 3 citations
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics #math.PR #math.ST #stat.TH

paper · pdf · doi:10.48550/arxiv.1206.6951

arxiv created 2012/06/29 · openalex publication_date 2012/06/29 · arxiv updated 2012/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper explores a conditional Gibbs theorem for a random walkinduced by i.i.d. (X₁,..,X_n) conditioned on an extreme deviation of its sum (S₁ⁿ=na_n) or (S₁ⁿ>na_n) where a_n→∞. It is proved that when the summands have light tails with some additional regulatity property, then the asymptotic conditional distribution of X₁ can be approximated in variation norm by the tilted distribution at point a_n, extending therefore the classical LDP case.

Citations

• Stretched random walks and the behaviour of their summands

Cited by

• An Abelian theorem with application to the conditional Gibbs principle
• Light tails: Gibbs conditional principle under extreme deviation
• A Gibbs Conditional theorem under extreme deviation

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