# #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 $$mathbb E_pi[g(X_1)]. I am curious about what we can say about $$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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