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The aim of this note is to give the exact asymptotics ofPsups [0,T]X(s)>uasu->~,where {X(t):t>=0} is a centered Gaussian process with stationary increments and T is an independent non-negative random variable with regularly varying tail distribution. In addition, we obtain explicit lower and upper bounds for the prefactor. As an example we analyze the case of X(t) being a fractional Brownian...
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