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$X_k$'s are i.i.d. Suppose $X_k$ is symmetric and $E[|X_k|^{3/4}]<\infty$. Do we have $S_n/n \rightarrow 0$ either in probability or almost surely, where $S_n$ is the partial sum.

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That's a curious moment condition. Where did you get that from? –  cardinal Nov 13 '11 at 18:51
    
basically a random one < 1. –  epsilon Nov 13 '11 at 18:56

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up vote 4 down vote accepted

No.

Take the standard Cauchy distribution: $$\newcommand{\rd}{\,\mathrm{d}} \mathbb E |X|^{3/4} = 2 \int_{0}^{\infty} \frac{x^{3/4}}{\pi (1+x^2)} \rd x= \csc(\pi/8) < \infty \>, $$ but $S_n/n$ is equal in distribution to $X$. That is, it is also standard Cauchy no matter how large $n$ becomes.

More generally, note that this same counterexample works for any $0 < p < 1$ since $$ \frac{\pi}{2} \mathbb E|X|^p = \int_{0}^\infty \frac{x^p}{1+x^2} \rd x \leq 1 + \int_{1}^\infty x^{p-2} \rd x = \frac{2-p}{1-p} < \infty \> . $$

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Thank you. It's a nice example –  epsilon Nov 13 '11 at 19:19
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Thanks. The Cauchy serves as a counterexample for many statements regarding probability and mathematical statistics. Usually, at least in a setting that admits continuous distributions, it's a good one to try out on any new conjecture or "result". –  cardinal Nov 13 '11 at 19:27

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