TY - JOUR

T1 - An almost sure functional limit theorem at zero for a class of Lévy processes normed by the square root function, and applications

AU - Buchmann, Boris

AU - Maller, Ross

AU - Szimayer, Alex

PY - 2008/9

Y1 - 2008/9

N2 - A recent result of Bertoin, Doney and Maller (Ann. Prob., 2007) gives an integral condition to characterize the class of Lévy processes X(t) for which lim supt↓0 |X (t)|/√t ∈ (0, ∞) occurs almost surely (a.s.). For such processes we have a kind of almost sure "iterated logarithm" result, but without the logs. In the present paper we prove a functional version of this result, which then opens the way to various interesting applications obtained via a continuous mapping theorem. We set these out in a rigorous framework, including a characterisation of the existence of an a.s. cluster set for the interpolated process, appropriate to the continuous time situation. The applications relate to functional laws for the supremum, reflected and a variety of other processes, including a class of stochastic differential equations, where we aim to give as informative a description as we can of the functional limit sets.

AB - A recent result of Bertoin, Doney and Maller (Ann. Prob., 2007) gives an integral condition to characterize the class of Lévy processes X(t) for which lim supt↓0 |X (t)|/√t ∈ (0, ∞) occurs almost surely (a.s.). For such processes we have a kind of almost sure "iterated logarithm" result, but without the logs. In the present paper we prove a functional version of this result, which then opens the way to various interesting applications obtained via a continuous mapping theorem. We set these out in a rigorous framework, including a characterisation of the existence of an a.s. cluster set for the interpolated process, appropriate to the continuous time situation. The applications relate to functional laws for the supremum, reflected and a variety of other processes, including a class of stochastic differential equations, where we aim to give as informative a description as we can of the functional limit sets.

KW - Almost sure convergence

KW - Iterated logarithm laws

KW - Local behaviour

KW - Lévy process

KW - Strassen's functional LIL

UR - http://www.scopus.com/inward/record.url?scp=45149119635&partnerID=8YFLogxK

U2 - 10.1007/s00440-007-0103-0

DO - 10.1007/s00440-007-0103-0

M3 - Article

SN - 0178-8051

VL - 142

SP - 219

EP - 247

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 1-2

ER -