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 $φ(t):=\log \mathbb E e^{t X_k}<\infty$ for all $t>-σ_0$ and some $σ_0>0$. 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 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 $ψ(t):=2φ(t)/t^2$ 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 $d_*=1/φ(t_*)>0$. 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.