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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?

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  • $\begingroup$ Is $S_n$ different from the sample mean of the $n$ elements? $\endgroup$
    – Dave
    Sep 8 at 18:33
  • $\begingroup$ Hi, $S_n$ is the summation of $n$ i.i.d. random variables, $S_n/n$ is the empirical mean of $X_1, ... X_n$. $\endgroup$
    – Bravo
    Sep 8 at 18:35
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    $\begingroup$ Check the alpha-stable keyword. $\endgroup$
    – Xi'an
    Sep 8 at 18:50
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    $\begingroup$ "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. $\endgroup$ Sep 15 at 6:35
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    $\begingroup$ 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). $\endgroup$ Sep 15 at 6:51

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