Problem of the Week #278 - Sep 11, 2018

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Euge

MHB Global Moderator
Staff member
Here is this week's POTW:

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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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Note $$T_n = \exp\left(\frac{1}{n}\sum_{i = 1}^n \ln X_n\right)$$ and by the weak law of large numbers, $$\frac{1}{n}\sum_{i = 1}^n\ln X_i \rightarrow \mu\quad \text{in probability}$$ Since, in addition, the map $x\mapsto e^x$ is continuous, then $T_n \to e^\mu$ in probability.