A refinement of asymptotics in the Prokhorov–Donsker invariance principle for integral functionals

An asymptotic expansion with order of accuracy $o(1/\sqrt n)$ is constructed for the distribution function (d.f.) of an integral functional of a random walk $S_n(t)$. The first stage of the proof consists of an approximation (in a certain sense) of a stochastic process distribution by that of a generalized Poisson process $\pi_n(t)$ with $o(1/\sqrt n)$ accuracy. The second stage is an investigation of d.f. asymptotics for integral functionals $\pi_n(t)$. An asymptotic expansion with $o(n^{-3/2})$ accuracy is also constructed.