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The behavior near the origin of the supremum functional in a process with stationary independent increments
Published online by Cambridge University Press: 14 July 2016
Abstract
Let {X(t),t ≧0} be a process with stationary independent increments which is stochastically continuous with right-continuous paths and normalized so that X(0)=0. Let Z1(t) = X(t), Z2(t) = sup0≦s≦tX(s) and Z3 (t) = largest positive jump of X in (0, t] if there is one; = 0 otherwise. Then for i = 1,2,3 and x > 0: limt↓0t—1P[Zi(t) > x] = M+(x) at all points of continuity of M+, the Lévy measure of X.
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- Copyright © Applied Probability Trust 1975
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