Limiting distribution for the maximal standardized increment of a random walk

Let $X_1,X_2,\mathrm{.}\mathrm{.}\mathrm{.}$ be independent identically distributed random variables with $\mathbb{E}{X}_k=0$, $\mathrm{V}\mathrm{a}\mathrm{r}{X}_k=1$. Suppose that $\varphi(t):=\log \mathbb E e^{t X_k}<\infty$forall$t>-\sigma_0$andsome$\sigma_0>0$\mathrm{.}Let$S_k=X_1+…+X_k$and$S_0=0$. We are interested in the limiting distribution of the multiscale scan statistic $$M_n=\underset{0\le i<j\le n}{\max }\frac{S_j-S_i}{\sqrt{j-i}}\mathrm{.}$$ 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 $\phi (t)<t^2\mathrm{/}2$ 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 $p=q\mathrm{/}(q-2)$ and $q\in 3,4,\mathrm{.}\mathrm{.}\mathrm{.}$ 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 $\psi (t):=2\phi (t)\mathrm{/}{t}^2$ attains its maximum $m_{\ast }>1$ at some unique point $t=t_{\ast }\in (0,\mathrm{\infty })$. 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 $\exp \{-x^{2-\epsilon }\}$, 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.