Normal approximation for $U$- and $V$-statistics of a stationary absolutely regular sequence

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 sufficient conditions for the asymptotic normality (for $n\to\infty$ and common centering and normalization) of the distribution of the nonhomogeneous $U$-statistic of order $r$ which is given on the sequence $X_{n,1},\ldots,X_{n,n}$ with a kernel also dependent on $n$. The same results for $V$-statistics also hold. To analyze sums of dependent random variables with rare strong dependencies, the proof uses the approach that was proposed by S. Janson in 1988 and upgraded by V. Mikhailov in 1991 and M. Tikhomirova and V. Chistyakov in 2015.