Exact small ball asymptotics in $L_2$-norm for finite dimensional perturbations of Gaussian processes: a spectral approach

We consider the problem of small ball behavior in $L_2$-norm for some finite dimensional perturbations of Gaussian processes. In noncritical case the explicit relation between exact small ball asymptotic behaviors for initial and perturbed processes is obtained. In critical case, if the perturbation is “good” enough, we can obtain a similar result; otherwise, theorems of a general type cannot be proved. In addition, we consider some natural examples from statistics (processes of Durbin), which appear to be critical in our sense. For them, general theorems do not work, but we develop technique that simplifies calculations in each particular case.