Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 26G / c / i / Solution

By the union bound,For an $L^2$ random variable, $t^2\mathbb{P}(|X_1|>t)\to 0$. Taking $t=\epsilon \sqrt{n}$ makes the right side tend to zero, proving $M_n/\sqrt{n}\to 0$ in probability.

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