I have a question concerning the proof of theorem 2.5 at page 395 of the paper Noncommuting Random Products, by H. Furstenberg, Trans. Amer. Math. Soc., 1963.
The statement is as follows:
Let $\mu$ be a probability measure on a group $G$. Let $M$ be a locally compact $G$-space, which does not admit a stationary measure for $\mu$, that is, there is no $\nu$ on $M$ such that $\mu*\nu=\nu$. Let $Z_n$ be a $\mu$-process on $M$, that is $Z_i$ are $M$ valued random variables such that the conditional distribution of $Z_{n+1}$ given $Z_{n},\dots,Z_0$ is $\mu*\delta_{Z_n}$. Let $\Delta\subset M$ be compact. Define $n(k)$ as the index $n$ for which $Z_n\in\Delta$ for the $k$-th time. Then $n(k)/k\to\infty$ with probability $1$.
The proof starts with stating that it is enough to show that for each compactly supported $\psi$ we have that with probability $1$ \begin{equation} \frac{1}{n}\sum_0^{n-1}\psi(Z_k)\to 0 \end{equation} This I do understand. However, Furstenberg proceeds to say that if this formula does not hold for some $\psi'$, then there exists a subsequence $n_i$ such that \begin{equation} \frac{1}{n_i}\sum_0^{n_i-1}\psi'(Z_k)\to \alpha\neq 0, \end{equation} and he seems to assume in what follows that neither $n_i$ nor $\alpha$ are random.
** Question: ** How is it that the second equation is the negation of the first? Is Furstenberg implicitly using here some "strong law of large numbers"?