Limiting distribution for the maximal standardized increment of a random walk

Abstract

Let $X_1,X_2,...$ be independent identically distributed random variables with $\mathbb{E}{X}_k=0$, $\mathrm{Var}{X}_k=1$. Suppose that $\phi (t):=\log \mathbb{E}{e}^{t{X}_k}<\infty$ for all $t>-{\sigma }_0$ and some ${\sigma }_0>0$. Let $S_k=X_1+...+X_k$ and $S_0=0$. We are interested in the limiting distribution of the multiscale scan statistic $$M_n=\underset{0\le i<j\le n}{\max }\frac{S_j-S_i}{\sqrt{j-i}}.$$ We prove that for an appropriate normalizing sequence $a_n$, the random variable $M_n^2-a_n$ converges to the Gumbel extreme-value law $\exp \{-e^{-cx}\}$. The behavior of $M_n$ depends strongly on the distribution of the $X_k$'s. We distinguish between four cases. In the superlogarithmic case we assume that $\phi (t)<t^2/2$ for every $t>0$. In this case, we show that the main contribution to $M_n$ comes from the intervals $(i,j)$ having length $l:=j-i$ of order $a(\log n{)}^p$, $a>0$, where $p=q/(q-2)$ and $q\in 3,4,...$ is the order of the first non-vanishing cumulant of $X_1$ (not counting the variance). In the logarithmic case we assume that the function $\psi (t):=2\phi (t)/t^2$ attains its maximum $m_{\ast }>1$ at some unique point $t=t_{\ast }\in (0,\infty )$. In this case, we show that the main contribution to $M_n$ comes from the intervals $(i,j)$ of length $d_{\ast }\log n+a\sqrt{\log n}$, $a\in \mathbb{R}$, where $d_{\ast }=1/\phi (t_{\ast })>0$. In the sublogarithmic case we assume that the tail of $X_k$ is heavier than $\exp \{-x^{2-\epsilon }\}$, for some $\epsilon >0$. In this case, the main contribution to $M_n$ comes from the intervals of length $o(\log n)$ and in fact, under regularity assumptions, from the intervals of length $1$. In the remaining, fourth case, the $X_k$'s are Gaussian. This case has been studied earlier in the literature. The main contribution comes from intervals of length $a\log n$, $a>0$. We argue that our results cover most interesting distributions with light tails.

44 pages, 6 figures

References in corpus (2)

• Extreme-Value Analysis of Standardized Gaussian Increments
• The limit distribution of the maximum increment of a random walk with regularly varying jump size distribution