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An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments. (English) Zbl 0607.60031
Let X(t), \(t\geq 0\), be a Gaussian process with mean 0 and stationary increments. If the incremental variance function \(\sigma^ 2(t)\) is convex and \(\sigma^ 2(t)=o(t)\) for \(t\to 0\), then \[ P(\max_{[0,t]}X(s)>u)\sim P(X(t)>u) \] for \(u\to \infty\) and each \(t>0\). This should be compared with Brownian motion where \(\sigma^ 2(t)=t\) and P(\(\max_{s\leq t}X(s)>u)=2P(X(t)>u)\).
Reviewer: J.Cuzick

MSC:
60G15 Gaussian processes
60G17 Sample path properties
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