L2-Small Ball Asymptotics for a Family of Finite-Dimensional Perturbations of Gaussian Functions.

In the present paper, we study the small ball probabilities in L2-norm for a family of finite-dimensional perturbations of Gaussian functions. We define three types of perturbations: noncritical, partially critical and critical; and derive small ball asymptotics for the perturbated process in terms of the small ball asymptotics for the original process. The natural examples of such perturbations appear in statistics in the study of empirical processes with estimated parameters (the so-called Durbin's processes). We show that the Durbin's processes are critical perturbations of the Brownian bridge. Under some additional assumptions, general results can be simplified. As an example, we find the exact L2-small ball asymptotics for critical perturbations of the Green processes (the processes whose covariance function is the Green function of the ordinary differential operator). [ABSTRACT FROM AUTHOR]