Assume $Z_1, Z_2, Z_3,...$ are independent and identically distributed R.V.s s.t. $Z_n∈(-1,1)$. Prove the following:
1) $\prod_{n=1}^i Z_n \to 0$ as $n \to \infty$. almost surely.
2) $\prod_{n=1}^i Z_n \to 0$ as $n \to \infty$. in $L_1$.
for $L_1$, since its $L_1$, we have to prove the expectation of product of all $Z_n$ converges to 0; since independent and identically distributed, it is obvious that its the product of all $E(Z_n)$ and $|E(Z_n)|$ should be less than $1$, but how would i prove $|E(Z_n)|<1$? if i have that, then $|E(Z_n)|^n \to 0$ as $n \to \infty$
for (1), im not too sure... do i fix a w s.t. $|X_n(w)|<1$ for sure, $|X_n(w)|^n\to 0$ for all $w$ then. Is this right? I feel like its like tooo similar to the proof as $L1$.
First if $E[|Z|] \ge 1$ then $E[1-|Z|] \le 0$. Or $1-|Z| \ge 0$ so $|Z| = 1$ a.s.. This proves that $E[|Z|] < 1$ and after you will have the convergence to $0$ in $L_1$.
Now for the first statement, we have $E[\ln|Z|] < 0$. Indeed, if $E[\ln|Z|] \ge 0$, then and because $\ln|Z| \le 0$ we have $\ln|Z| = 0$ a.s. and so $|Z| = 1$ a.s.
Using the law of large numbers $$\frac{1}{n} \sum_{k=1}^n \ln|Z_k| \rightarrow \ln|Z| < 0 \text{ a.s}$$ and so $$\sum_{k=1}^n \ln|Z_k| \rightarrow -\infty \text{ a.s}$$ this proves that $$\prod_{k=1}^n |Z_k| \rightarrow 0$$