Extreme-Value Analysis of Standardized Gaussian Increments

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 $$L_n=\underset{0\le i<j\le n}{\max }\frac{S_j-S_i}{\sqrt{j-i}}.$$ 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.

37 pages

References in corpus (2)

• On Lerch's transcendent and the Gaussian random walk
• Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices

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