Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 201 / 2 / b / Solution

After replacing $X_j$ by $X_j-\mu$, the maximal inequality for independent averages givesThis follows from the Doob Lp maximal inequality by dyadically grouping the partial sums. The strong law makesalmost surely. The displayed maximal function is in $L^p$, so dominated convergence applied to its $p$th power proves convergence in $L^p$.