## Abstract

Let ${({X}_{i})}_{i\ge 1}$ be i.i.d. random variables with $\mathsf{E}\phantom{\rule{0.1667em}{0ex}}{X}_{1}=0$, 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 ${T}_{n}(\mathit{\gamma})\phantom{\rule{-0.1667em}{0ex}}=\phantom{\rule{-0.1667em}{0ex}}{max}_{0\le j<k\le n}|{X}_{j+1}+\cdots +{X}_{k}|{(k-j)}^{-\mathit{\gamma}}$ in the threshold case ${\mathit{\gamma}}_{a}\phantom{\rule{-0.1667em}{0ex}}:=\phantom{\rule{-0.1667em}{0ex}}1\u22152-1\u2215a$ 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 ${\mathit{\sigma}}^{2}=\mathsf{E}\phantom{\rule{0.1667em}{0ex}}{X}_{1}^{2}$, $T(\mathit{\gamma})$ 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 ${c}_{n}={n}^{-1\u2215a}$ 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 ${V}_{a,\mathit{\sigma},\mathit{\tau}}$.

## Citation

Alfredas Račkauskas. Charles Suquet. "On the asymptotic of the maximal weighted increment of a random walk with regularly varying jumps: the boundary case." Electron. J. Probab. 26 1 - 31, 2021. https://doi.org/10.1214/21-EJP691

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