Assume that $\displaystyle X_1,X_2,...,X_n$ are iid uniform on [0,1]. Show that $\displaystyle (X_1X_2...X_n)^\frac{1}{n}\to e^{-1}$ a.e. for $\displaystyle n\to\infty$.
Assume that $\displaystyle X_1,X_2,...,X_n$ are iid uniform on [0,1]. Show that $\displaystyle (X_1X_2...X_n)^\frac{1}{n}\to e^{-1}$ a.e. for $\displaystyle n\to\infty$.
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Express the power using $\displaystyle a^b=e^{b\log a}$ and use the law of large numbers.
(Or equivalently take the logarithm of the sequence and apply the law of large numbers)