On the Asymptotic Power of the Tests of Fit by Near Alternatives

Let $G_n^*(u)$ be the empirical distribution of a sample of size $n$ from a distribution function $G(u)$, $0 \leqq u\leqq 1$, and $\beta_n (u)=\sqrt n(G_n^*(u)-u)$. It is proved, that if $G(u)=G_n(u)$ and $\sqrt n(G_n (u)-u)\to\delta(u)$ as $n\to\infty$, $\beta_n(u)$ converges to $\beta(u)+\delta(u)$, where $\beta(u)$ is the gaussian process with ${\mathbf M}\beta(u)=0$, ${\mathbf M}\beta(u)\beta (v)=\min(u,v)-uv$. The exact definitions of convergence are indicated in the statements of theorems.