#StackBounty: #probability #distributions #expected-value #calculus When will $mathbb{E}[g(S_n/n)]$ exist given $mathbb{E}[g(X_1)]$ e…

Bounty: 50

Suppose $X_1, X_2,…, X_n$ are i.i.d. random variables with distribution $pi$ on some probability space. Let $g$ be a measurable function such that $mathbb E_pi[g(X_1)]<infty$. I am curious about what we can say about $mathbb E_pi[g(S_n/n)]$, where $S_n = sum_{k=1}^n X_k$?

My guess is the quantity $mathbb E_pi[g(S_n/n)]$ is not necessarily finite in general, but should be finite if $g$ satisfies appropriate regularity conditions. I am wondering if there are any existing studies on this topic?

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