Are successive recurrence times ergodic for an ergodic process?

Question

In paper "Successive Recurrence Times in a Stationary Process" by Shu-Teh Chen Moy (The Annals of Mathematical Statistics, Vol. 30, No. 4 (Dec., 1959), pp. 1254-1257), it was shown that for a stationary sequence of random variables $X_0,X_1,X_2...:\Omega\to\mathbb{X}$, successive recurrence times $\nu_1,\nu_2,\nu_3,...:\Omega\to\mathbb{N}$ of getting back to a set $B\subset\mathbb{X}$ (i.e., $X_{\nu_1+...+\nu_k}\in B$ and $X_i\not\in B$ for other $i$) form also a stationary process, albeit with the probability measure of process $X_0,X_1,X_2,...$ conditioned on event $(X_0\in B)$. In turn, this statement leads to a generalization of the famous Kac lemma $E(\nu_k|X_0\in B)=1/P(X_0\in B)$, proven by Mark Kac for $k=1$. It is tempting to ask whether process $\nu_1,\nu_2,\nu_3,...$ is also ergodic if process $X_0,X_1,X_2,...$ is ergodic. Note that the shift operations in these two processes are different. Has this problem been solved or attacked? Are there counterexamples? It seems quite a natural question. I know little of ergodic theory.