everyone. I have only recently embarked on a journey through measure theoretic probability and was pondering the following question:
Consider a process with $$X_n|X_{n-1} \sim N(0, X_{n-1}^2).$$ Define $S_n = X_1 + X_2 + \cdots + X_n$ and $T_n = X_1^2 + X_2^2 + \cdots + X_n^2.$ Do $\frac{S_n}{n}$ and $\frac{T_n}{n}$ converge almost surely to $0$? If so, why? From my simulations it seems that both certainly converge to 0, though I can't seem to justify why.