An almost sure functional limit theorem at zero for a class of Lévy processes normed by the square root function, and applications
type: Item
instance of: scholarly article
title: An almost sure functional limit theorem at zero for a class of Lévy processes normed by the square root function, and applications
zbMATH Open document ID: 1153.60025
author: Boris Buchmann
author: Alex Szimayer
author: Ross A. Maller
publication date: 2008-10-16T00:00:00Z
review text: Let \(X\) be a Levy process with characteristic triple \((\gamma, \sigma^2=0, \Pi)\) and assume that \(V(x)=\int_{| y| \leq x} y^2 \Pi(dy) > 0\) for all \(x>0\). In a recent result of Bertoin, Doney and Maller it is stated that if \[ \lambda^*_X := \inf \{ a>0 : \int_0^1 \exp(-\frac{a^2}{2V(x)}) \frac{dx}{x} < \infty \} \in [0,\infty] , \] then almost sure \[ -\liminf_{t\downarrow 0} \frac{X(t)}{\sqrt t} = \limsup_{t\downarrow 0} \frac{X(t)}{\sqrt t} = \lambda^*_X . \] This in some way surprising result is supposed to hold in the present paper: let \(\lambda^*_X \in (0,\infty)\). Then the authors give a functional version (Theorem 2) of the statement above. Using a kind of continuous mapping theorem (Theorem 3), they characterize the almost sure cluster sets for a variety of random processes under which: (i) the interpolated process \(Z^X_t(y) := \frac{X(ty)}{\lambda^*_X \sqrt t} , \;0\leq y\leq 1 \); (ii) the reflected processes \(R(t):= (\sup_{s\leq t}X(s)) - X(t) , \;r(t):= X(t) - (\inf_{s\leq t}X(s)) \); (iii) the process \(L\) satisfying \(L(u)= \int_0^u B(L(t))\,dt + X(u) , \;0\leq u \leq 1\), where \(B\) is a Lipschitz function. Rigorous proofs of all statements are given in the last three sections.
reviewed by: Elisaveta Pancheva
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 5351827
zbMATH Keywords: Levy processes
zbMATH Keywords: local behaviour
zbMATH Keywords: almost sure convergence
MaRDI profile type: Publication
full work available at URL: s00440-007-0103-0
OpenAlex ID: W2157737860
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DOI: 10.1007/S00440-007-0103-0
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 5351827