On asymptotic normality of U-statistic of a stationary absolutely regular sequence in a triangular array scheme.

Let $(X_{n,t})_{t=1}^{\infty}$ be a stationary absolutely regular sequence of real random variables with the distribution dependent on the number~$n$. The paper presents the sufficient conditions for the asymptotic normality (for $n\to\infty$ and common centering and normalization) of the distribution of the second-order $U$-statistic of $X_{n,1},\ldots,X_{n,n}$ with a kernel also dependent on $n$. To analyze sums of dependent random variables with rare strong dependencies the proof uses the approach that was proposed by Svante Janson in 1988 and upgraded by Mikhailov in 1991 and Tikhomirova and Chistyakov in 2015.