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Suppose that $(X_k)_{k \in \mathbb{N}}$ is a sequence of independent and identically distributed random variables such that $X_1$ has density function $f_{X_1}(x) = \frac{1}{\pi(1+x^2)}$, $x \in \mathbb{R}$, i.e., it is Cauchy distributed.

Let $Y_n = \max(X_1, X_2, \cdot\cdot\cdot, X_n)$. I would to know how can we prove that $$ \lim_{n \rightarrow \infty} Y_n = \infty \text{ almost surely}. $$

Thanks very much.

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    $\begingroup$ The specifics of the situation are irrelevant, the result uses only that $P(X_1\geqslant x)\ne0$ for every $x$. $\endgroup$
    – Did
    Nov 24, 2014 at 19:58

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Notice that for each $R$, $\mathbb P\{Y_n\leqslant R\}=(\mathbb P\{X_1\leqslant R\})^n$. Since the sequence of sets $(\{Y_n\leqslant R\})$ is non-increasing, we have $$\mathbb P\left\{\sup_kX_k \leqslant R\right\}=\lim_{n\to\infty} (\mathbb P\{X_1\leqslant R\})^n.$$ We thus have that $\mathbb P\left\{\sup_kX_k \leqslant R\right\}=0$ for each $R$ as long as $\mathbb P\{X_1\leqslant R\}\lt 1$ for each positive $R$.

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