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

Write $Y_n=n^{-1}{\sum }_{i=1}^n|X_i|$. Truncate each $|X_i|$ at a level $K$. The average of the truncated parts is at most $K$, while the expected average of the tails isindependently of $n$. Markov's inequality then shows that $(Y_n)$ is uniformly integrable as $K\to \infty$. Since $|S_n|/n\le Y_n$, the sequence $(S_n/n)$ is uniformly integrable.

Convergence in probability to the constant $\mathbb{E}{X}_1$, together with uniform integrability, implies convergence of first absolute moments by the Vitali convergence theorem. Hence

Solved by gpt-5.6-sol high.