In this talk, we show the maximally selected standardized U-statistics based on an absolutely regular process converges in distribution to an infinite sum of weighted chi-square random variables in the degenerate case. The result generalizes that of Horv'ath and Shao (1996) as follows.

{\bf Theorem A.}
Let $X_1, X_2, \cdots$ be i.i.d. random variables with common distribution function $F(x)$ defined on a probability space $(\Omega, {\bf F}, P)$. Let $h(x,y)$ be a symmetric function and define
\begin{eqnarray}
U_{k,n} ^* \ = \ \sum_{i=1} ^k \sum_{j=k+1} ^n h(X_i, X_j) - k(n-k) \theta \nonumber
\end{eqnarray}
where
\begin{eqnarray}
\theta \ = \ E h(X_1, X_2) . \nonumber
\end{eqnarray}
{\it We assume that
\begin{eqnarray}
E \tilde{h}^2(X_1) \ = \ 0 \quad {\rm and} \quad 0 \ < \sigma_0 ^2 \ = \
E ( h(X_1, X_2) - \theta)^2 \ < \infty . \nonumber
\end{eqnarray}
Then, as} $n \to \infty$
\begin{eqnarray}
(2 \log \log n)^{-\frac{1}{'day':2}} \max_{1 \le k < n} \frac{|U_{k,n} ^* |}{\sqrt{k(n-k)}} \ \mathop{\to}^D \ Z .
\end{eqnarray}