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Let $\{X_n\}_{n \geq 1}$ be a sequence of iid random variables with pdf $$f(x) = \begin{cases} \frac {c} {x^2 \log |x|}, & |x| \gt 2, \\ 0, & \text {otherwise}, \end{cases}$$ where $c$ is a normalizing constant. Let $S_n = \sum\limits_{k=1}^{n} X_k.$ Show that $\frac {S_n} {n} \xrightarrow {p} 0.$

Let us take $\varepsilon \gt 0.$ We need to show that $\mathbb P \left [\left \lvert \frac {S_n} {n} \right \rvert \gt \varepsilon \right ] \to 0$ as $n \to \infty.$ It is clear that $X_n$'s have infinite mean and hence we cannot apply WLLN to conclude the result. By rough estimates we get

$$\mathbb P \left [\left \lvert \frac {S_n} {n} \right \rvert \gt \varepsilon \right ] \leq n\ \mathbb P [|X_1| \gt \varepsilon]$$ which also does not seem to help much. Any idea?

Anacardium
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  • Isn't $\mathbb E[X_i] = 0$ due to symmetry? If so, I think WLLN directly gives the statement. – Vezen BU Feb 15 '24 at 03:56
  • @Vezen BU$:$ In order to apply WLLN don't we need $\mathbb E [|X_1|] \lt \infty$ i.e. $X_1 \in L^1\ $? – Anacardium Feb 15 '24 at 08:24
  • Would the result here be applicable? https://math.stackexchange.com/questions/1644218/slln-when-the-expectation-in-infinite – Vezen BU Feb 15 '24 at 13:34

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