M 511

I consider the problem of small ball probabilities for Gaussian processes in $L_2$-norm. I focus on the processes which are important in statistics (e.g. Kac-Kiefer-Wolfowitz processes), which are finite dimentional perturbations of Gaussian processes. Depending on the properties of the kernel and perturbation matrix I consider two cases: non-critical and critical. * For non-critical case I prove the general theorem for precise asymptotics of small deviations. * For a huge class of critical processes I prove a general theorem in the same spirit as for non-critical processes, but technically much more difficult. * At the same time a lot of processes naturally appearing in statistics (e.g. Durbin, detrended processes) are not covered by those general theorems, so I treat them separately using methods of spectral theory and complex analysis.

Yulia Petrova

St. Petersburg