On the asymptotic of the maximal weighted increment of a random walk with regularly varying jumps: the boundary case

Abstract

Let ${(X_i)}_{i\ge 1}$ be i.i.d. random variables with $\mathsf{E}{X}_1=0$, regularly varying with exponent $a>2$ and $t^{a}P(|X_1|>t)\sim L(t)$ slowly varying as $t\to \mathrm{\infty }$. We give the limit distribution of $T_n(\mathit{\gamma })={max}_{0\le j<k\le n}|X_{j+1}+\cdots +X_k|{(k-j)}^{-\mathit{\gamma }}$ in the threshold case ${\mathit{\gamma }}_a:=1∕2-1∕a$ which separates the Brownian phase corresponding to $0\le \mathit{\gamma }<{\mathit{\gamma }}_a$ where the limit of $T_n(\mathit{\gamma })$ is $\mathit{\sigma }T(\mathit{\gamma })$, with ${\mathit{\sigma }}^2=\mathsf{E}{X}_1^2$, $T(\mathit{\gamma })$ is the γ-Hölder norm of a standard Brownian motion and the Fréchet phase corresponding to ${\mathit{\gamma }}_a<\mathit{\gamma }<1$ where the limit of $T_n(\mathit{\gamma })$ is $Y_a$ with Fréchet distribution $P(Y_a\le x)=exp(-x^{-a})$, $x>0$. We prove that $c_n^{-1}(T_n({\mathit{\gamma }}_a)-{\mathrm{\mu }}_n)$, converges in distribution to some random variable Z if and only if L has a limit ${\mathit{\tau }}^a\in [0,\mathrm{\infty }]$ at infinity. In such case, there are $A>0$, $B\in \mathbb{R}$ such that $Z=A{V}_{a,\mathit{\sigma },\mathit{\tau }}+B$ in distribution, where for $0<\mathit{\tau }<\mathrm{\infty }$, $V_{a,\mathit{\sigma },\mathit{\tau }}:=max(\mathit{\sigma }T({\mathit{\gamma }}_a),\mathit{\tau }{Y}_a)$ with $T({\mathit{\gamma }}_a)$ and $Y_a$ independent and $V_{a,\mathit{\sigma },0}:=\mathit{\sigma }T({\mathit{\gamma }}_a)$, $V_{a,\mathit{\sigma },\mathrm{\infty }}:=Y_a$. When $\mathit{\tau }<\mathrm{\infty }$, a possible choice for the normalization is $c_n=n^{-1∕a}$ and ${\mathrm{\mu }}_n=0$, with $Z=V_{a,\mathit{\sigma },\mathit{\tau }}$. We also build an example where L has no limit at infinity and ${(T_n(\mathit{\gamma }))}_{n\ge 1}$ has for each $\mathit{\tau }\in [0,\mathrm{\infty }]$ a subsequence converging after normalization to $V_{a,\mathit{\sigma },\mathit{\tau }}$.