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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?

mhdadk
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Bravo
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  • Is $S_n$ different from the sample mean of the $n$ elements? – Dave Sep 08 '21 at 18:33
  • Hi, $S_n$ is the summation of $n$ i.i.d. random variables, $S_n/n$ is the empirical mean of $X_1, ... X_n$. – Bravo Sep 08 '21 at 18:35
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    Check the `alpha-stable` keyword. – Xi'an Sep 08 '21 at 18:50
  • The conditions must apply to $g$ and $\pi$ simultaneously. Regardless, fairly mild conditions will assure the second expectation is finite. For instance, it suffices for $|g|$ to be bounded above by some positive convex function with finite expectation. – whuber Sep 08 '21 at 19:26
  • Is $X_i$ continuous? Otherwise, $\frac{S_n}{n}$ may not be in the domain of $g$. – krkeane Sep 14 '21 at 16:15
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    *"My guess is the quantity $\mathbb E_\pi[g(S_n/n)]$ is not necessarily finite in general..."* we can be sure about this. Consider as an example $g(x) = 1/x$ and $X$ distributed according to a density of zero at $x=0$ and a mean equal to zero. – Sextus Empiricus Sep 15 '21 at 06:35
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    Another way that breaks the generality are variables whose mean is distributed with a greater spread, ie stochastically larger. E.g. the average of $n$ Levy distributed variables is distributed a single Levy distributed variable multiplied by $n$. I can also imagine that heavy tail distributions will have such behaviour (and even more extreme) where the mean is stochastically larger, but it is difficult to describe because they are not stable distributions (possibly you could use the maximum from the sample divided by n as an estimate). – Sextus Empiricus Sep 15 '21 at 06:51

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