Extreme-Value Analysis of Standardized Gaussian Increments

TL;DR

This paper investigates the asymptotic distribution of the maximum standardized partial sums of i.i.d. Gaussian variables, showing convergence to a Gumbel distribution and relating it to maxima of dependent Gaussian variables.

Contribution

It establishes the limiting distribution of the maximum standardized partial sums for Gaussian sequences and extends the results to Brownian motion, revealing dependence effects.

Findings

Distribution of $L_n$ converges to Gumbel distribution

$L_n$ behaves like maximum of $Hn \log n$ independent Gaussians

Results extend to Brownian motion case

Abstract

Let $\{X_i,i=1,2,...\}$ be i.i.d. standard gaussian variables. Let $S_n=X_1+...+X_n$ be the sequence of partial sums and $$ L_n=\max_{0\leq i<j\leq n}\frac{S_j-S_i}{\sqrt{j-i}}. $$ We show that the distribution of $L_n$, appropriately normalized, converges as $n\to\infty$ to the Gumbel distribution. In some sense, the the random variable $L_n$, being the maximum of $n(n+1)/2$ dependent standard gaussian variables, behaves like the maximum of $Hn \log n$ independent standard gaussian variables. Here, $H\in (0,\infty)$ is some constant. We also prove a version of the above result for the Brownian motion.