The distribution of Sherman's weighted statistic for contiguous alternatives

Let $U_n(1),\dots,U_n(n)$ be the variational series of a simple random sample of size $n$ from the uniform distribution on [0, 1].
In this paper, the asymptotical distribution (as $n\to\infty$) of statistic
$$ \xi_n=\frac{1}{2}\sum_{j=1}^{n+1}a\biggl(\frac{j}{n+1}\biggr) \biggl|\varphi_n(U_n(j))-\varphi_n(U_n(j-1))-\frac{1}{n+1}\biggr| $$
is derived, where $a(u)$, $0\le u\le 1$, is a weight function,
$$ \varphi_n(u)=u+\frac{1}{\sqrt{n+1}}\int_0^u b_n(x)\,dx,\qquad\int_0^u b_n(x)\,dx=O(1). $$

The result obtained is used to construct a goodness-of-fit test.