Limiting distribution for the maximal standardized increment of a random walk

TL;DR

This paper investigates the limiting distribution of the maximum standardized increment of a random walk, revealing that it converges to a Gumbel law with behavior depending on the tail properties of the underlying distribution.

Contribution

The paper classifies the asymptotic behavior of the multiscale scan statistic based on different tail conditions of the distribution of the steps, extending previous results to a broader class of distributions.

Findings

In superlogarithmic case, main contribution from intervals of order $( ext{log } n)^p$.

In logarithmic case, contribution from intervals of length proportional to $ ext{log } n$.

In sublogarithmic case, contribution mainly from very short intervals, often of length 1.

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 $\varphi(t):=\log \mathbb E e^{t X_k}<\infty$ 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 $$ M_n=\max_{0\leq i <j\leq n} \frac{S_j-S_i}{\sqrt{j-i}}. $$ 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 $\exp\{-e^{-c x}\}$. 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 $\varphi(t)<t^2/2$ 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 $\psi(t):=2\varphi(t)/t^2$ attains its maximum $m_*>1$ at some unique point $t=t_*\in (0,\infty)$. In this case, we show that the main contribution to $M_n$ comes from the intervals $(i,j)$ of length $d_*\log n+a\sqrt{\log n}$, $a\in\mathbb R$, where $d_*=1/\varphi(t_*)>0$. In the sublogarithmic case we assume that the tail of $X_k$ is heavier than $\exp\{-x^{2-\varepsilon}\}$, for some $\varepsilon>0$. 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.