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- Jun 20, 2014

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Let $(X_n)_{n \in \Bbb N}$ be a sequence of positive i.i.d. random variables such that $E[\ln X_n]$ is a constant finite positive number $\mu$. Show that if $$T_n := \prod_{i = 1}^n X_i^{1/n}\quad (n = 1,2,3,...)$$ then $(T_n)_{n\in \Bbb N}$ converges in probability to $e^{\mu}$.

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