Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables from a \(\operatorname{CaUchy}(0,1)\) distribution. Prove that the mean of the distribution does not exist, and further prove that it can be shown that \(n P\left(\left|X_{1}ight|>night) ightarrow 2 \pi^{-1}\) as \(n ightarrow \infty\), so that the condition of Theorem 3.14 does not hold.

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Theorem 3.14. Let {X} be a sequence of independent random variables each having a common distribution F. Then Xn E(X18{|X1|; [0, n]}) > 0 as n if and only if - lim nP(Xn) = 0. n