On the preservation of statistical properties for subsequences of random sequences

Let $X$ be a one-sided sequence space and $\mathbf P$ be a probability measure on $X$ invariant under the shift transformation $T$ and such that the coordinates $x_i$ of $x\in X$ are weakly dependent. It is well-known that $\mathbf P$-almost every point $x\in X$ is $\mathbf P$-normal, i. e. for any sufficiently good $A\subset X$
$$ \lim_{n\to\infty}n^{-1}\operatorname{card}\{i:T^ix\in A,\ i\le n\}=\mathbf P(A). $$
We find conditions on the integer-valued sequence $\tau=(\tau_0,\tau_1,\dots)$ under which the normality of a point $x=(x_0,x_1,\dots)\in X$ s X implies that of the point $x'=(x_{\tau_0},x_{\tau_1},\dots)$.