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

Abstract

Let (Xi)i1 be i.i.d. random variables with EX1=0, regularly varying with exponent a>2 and taP(|X1|>t)L(t) slowly varying as t. We give the limit distribution of Tn(γ)=max0j<kn|Xj+1++Xk|(kj)γ in the threshold case γa:=121a which separates the Brownian phase corresponding to 0γ<γa where the limit of Tn(γ) is σT(γ), with σ2=EX12, T(γ) is the γ-Hölder norm of a standard Brownian motion and the Fréchet phase corresponding to γa<γ<1 where the limit of Tn(γ) is Ya with Fréchet distribution P(Yax)=exp(xa), x>0. We prove that cn1(Tn(γa)μn), converges in distribution to some random variable Z if and only if L has a limit τa[0,] at infinity. In such case, there are A>0, BR such that Z=AVa,σ,τ+B in distribution, where for 0<τ<, Va,σ,τ:=max(σT(γa),τYa) with T(γa) and Ya independent and Va,σ,0:=σT(γa), Va,σ,:=Ya. When τ<, a possible choice for the normalization is cn=n1a and μn=0, with Z=Va,σ,τ. We also build an example where L has no limit at infinity and (Tn(γ))n1 has for each τ[0,] a subsequence converging after normalization to Va,σ,τ.

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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

Information

Received: 30 March 2021; Accepted: 25 August 2021; Published: 2021
First available in Project Euclid: 23 September 2021

Digital Object Identifier: 10.1214/21-EJP691

Subjects:
Primary: 60G50, 60G70

JOURNAL ARTICLE
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Vol.26 • 2021
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