Large deviations of infinite intersections of events in Gaussian processes
Consider events of the form {Zs>=[zeta](s),s[set membership, variant]S}, where Z is a continuous Gaussian process with stationary increments, [zeta] is a function that belongs to the reproducing kernel Hilbert space R of process Z, and is compact. The main problem considered in this paper is identifying the function [beta]*[set membership, variant]R satisfying [beta]*(s)>=[zeta](s) on S and having minimal R-norm. The smoothness (mean square differentiability) of Z turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when [zeta](s)=s for s[set membership, variant][0,1] and Z is either a fractional Brownian motion or an integrated Ornstein-Uhlenbeck process.
Year of publication: |
2006
|
---|---|
Authors: | Mandjes, Michel ; Mannersalo, Petteri ; Norros, Ilkka ; van Uitert, Miranda |
Published in: |
Stochastic Processes and their Applications. - Elsevier, ISSN 0304-4149. - Vol. 116.2006, 9, p. 1269-1293
|
Publisher: |
Elsevier |
Keywords: | Sample-path large deviations Dominating point Reproducing kernel Hilbert space Minimum norm problem Fractional Brownian motion Busy period |
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