The limit distribution of the maximum increment of a random walk with dependent regularly varying jump sizes
type: Item
instance of: scholarly article
title: The limit distribution of the maximum increment of a random walk with dependent regularly varying jump sizes
zbMATH Open document ID: 1282.60053
author: T. Mikosch
author: Martin Moser
publication date: 2013-06-19T00:00:00Z
review text: The authors investigate the maximum increment of a random walk \((S_{n})\) starting with \(S_{0}=0\) with regularly varying jump sizes \((X_{t})\). The maximum increment of the random walk \((S_{n})\) is defined by the quantities \[ \tilde{M}_{n}^{(\gamma )}=\max_{1\leq l\leq n}\frac{1}{f(l)}\max_{0\leq k\leq n-l}\left| S_{k+l}-S_{k}\right| ,\qquad n\geq 1, \] and their centered analogs \[ \tilde{T}_{n}^{(\gamma )}=\max_{1\leq l\leq n}\frac{1}{f(l(1-l/n))} \max_{0\leq k\leq n-l}\left| S_{k+l}-S_{k}-l\frac{S_{n}}{n}\right| ,\qquad n\geq 1, \] where \(f\) is a non-decreasing function on \([0,\infty )\) satisfying \(f(1)=1\) and \(f(l)\geq l^{\gamma }\) for \(l\geq 1\) such that the relation \[ \lim_{n\rightarrow \infty }\inf_{1\leq l\leq d_{n}}\frac{f(l(1-l/n))}{f(l)}=1 \] holds for any increasing sequence \((d_{n})\) of positive numbers satisfying \( d_{n}^{2}/n\rightarrow 0\). The first author and \textit{A. Račkauskas} [Bernoulli 16, No. 4, 1016--1038 (2010; Zbl 1215.60018)] have previously studied the case of a random walk \((S_{n})\) with i.i.d. regularly varying Banach-valued jump sizes \((X_{t})\). Consider real-valued random variables \((X_{t})\), which are assumed to be regularly varying, that is, the generic element \(X\) of this sequence satisfies the tail balance condition \[ \operatorname{P}\left( X>x\right) =\tilde{p}x^{-\alpha }L(x)\qquad \text{and}\qquad \operatorname{P}\left( X\leq -x\right) =\tilde{q}x^{-\alpha }L(x),\qquad x\rightarrow \infty , \] where \(L\) is a slowly varying function, \(\alpha >0\) is the index of regular variation, and \(\tilde{p},\tilde{q}\geq 0\), \(\tilde{p}+\tilde{q}=1\). Assuming \(\operatorname{E}X=0\) if \(\operatorname{E}\left| X\right| <\infty \), their result states that, for \(\gamma >\max (0,\;0.5-\alpha ^{-1})\) and for the normalizing sequence \((a_{n})\) chosen according to \(n\operatorname{P}\left( \left| X\right| >a_{n}\right) \rightarrow 1\), \[ \lim_{x\rightarrow \infty }\operatorname{P}\left( a_{n}^{-1}\tild…
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 37M10
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 6176810
zbMATH Keywords: maximum increment of a random walk
zbMATH Keywords: dependent jump sizes
zbMATH Keywords: moving average process
zbMATH Keywords: GARCH process
zbMATH Keywords: stochastic volatility model
zbMATH Keywords: regular variation
zbMATH Keywords: extreme value distribution
reviewed by: Qihe Tang
MaRDI profile type: Publication
full work available at URL: s00440-012-0427-2
OpenAlex ID: W2034245644
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DOI: 10.1007/S00440-012-0427-2
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 6176810