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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
##### Keywords:
maxima; stationary increments
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