After replacing $X_j$ by $X_j-\mu$, the maximal inequality for independent averages gives
$$
\left\|\sup_{m\geq1}\frac{|S_m|}{m}\right\|_{L^p}
\leq C_p\|X_1\|_{L^p},
\qquad p>1.
$$
This follows from the <Doob Lp maximal inequality> by dyadically grouping the partial sums. The strong law makes
$$
\sup_{m\geq n}\frac{|S_m|}{m}\longrightarrow0
$$
almost surely. The displayed maximal function is in $L^p$, so <dominated convergence theorem>[dominated convergence] applied to its $p$th power proves convergence in $L^p$.

Solution

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

{c}
{title2=$L^p$}
{wiki=Doob's_martingale_inequality}

For $p>1$ and a martingale $(M_k)_{k\leq n}$,
$$
\left\|\max_{k\leq n}|M_k|\right\|_p
\leq\frac p{p-1}\|M_n\|_p.
$$


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