The limit distribution of the maximum increment of a random walk with regularly varying jump size distribution

TL;DR

This paper establishes that the maximum increment of a random walk with regularly varying jumps converges to the Fréchet distribution, providing insights into change point detection in stochastic processes.

Contribution

It proves the asymptotic distribution of the maximum increment for random walks with regularly varying jumps, extending results to Banach space-valued jumps.

Findings

Maximum increment converges to Fréchet distribution

Results apply to Banach space-valued jumps

Provides theoretical foundation for change point detection

Abstract

In this paper, we deal with the asymptotic distribution of the maximum increment of a random walk with a regularly varying jump size distribution. This problem is motivated by a long-standing problem on change point detection for epidemic alternatives. It turns out that the limit distribution of the maximum increment of the random walk is one of the classical extreme value distributions, the Fr\'{e}chet distribution. We prove the results in the general framework of point processes and for jump sizes taking values in a separable Banach space.