Large deviations of infinite intersections of events in Gaussian processes.

Abstract Consider events of the form {Zs≥ζ(s),sset membership, variantS}, where Z is a continuous Gaussian process with stationary increments, ζ is a function that belongs to the reproducing kernel Hilbert space R of process Z, and Click to view the MathML source is compact. The main problem considered in this paper is identifying the function β*set membership, variantR satisfying β*(s)≥ζ(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 ζ(s)=s for sset membership, variant[0,1] and Z is either a fractional Brownian motion or an integrated Ornstein-Uhlenbeck process.
Keywords: Sample-path large deviations; Dominating point; Reproducing kernel Hilbert space; Minimum norm problem; Fractional Brownian motion; Busy period
Mathematical subject codes: 60G15; 60K25; 60F10