zbMATH — the first resource for mathematics

Rough asymptotics of the probability of simultaneous high extrema of two Gaussian processes: the dual action functional. (English. Russian original) Zbl 1075.60026
Russ. Math. Surv. 60, No. 1, 167-168 (2005); translation from Usp. Mat. Nauk 60, No. 1, 171-172 (2005).
Let \(X(t)\), \(t\in T\), and \(Y(s)\), \(s\in S\), be two Gaussian processes with a.s. bounded trajectories and \(T\) and \(S\) be arbitrary sets. Then for any \(D\subset T\times S\), \[ \ln \mathbb{P}\left( \bigcup_{(t,s)\in D}\{ X(t)>u \}\cap \{ Y(t)>u \} \right) \sim -\frac{1}{2}I(X,Y;D)u^2,\quad u\to+\infty, \] where \(I\) is the minimum of the dual action functional for \(X\) and \(Y\).

60G15 Gaussian processes
60F10 Large deviations
60G70 Extreme value theory; extremal stochastic processes
Full Text: DOI