Exact $L_2$-small ball asymptotics for some Durbin processes

We find the exact $L_2$-small ball asymptotics for some Durbin processes. These processes are finite dimentional perturbations of the Brownian bridge $B(t)$ and naturally appear in statistics as limit ones when building goodness-of-fit tests of $\omega^2$-type for testing a sample for some distribution with estimated parameters. Earlier, in the work of Nazarov and Petrova, Kac–Kiefer–Wolfowitz processes (which correspond for testing normality) were considered, where a technique for obtaining asymptotics of oscillating integrals with a slowly varying amplitude was developed. Due to this, it is possible to calculate the asymptotics of small deviations for Durbin processes for certain distributions (Laplace, logistic, Gumbel, gamma).