Strong Law of Large Numbers under a Fourth Moment Bound

Let $(X_m)_{m\in\mathbb{N}}$ be a sequence of independent and identically distributed random variables on a probability space $(\Omega,\mathcal{F},P)$ such that $X_1^{4}$ has finite expectation (hence so do $X_1$, $X_1^{2}$, and $X_1^{3}$, since $|x|^{k}\le 1+x^{4}$ for $k\in\{1,2,3\}$ and every real $x$, and monotonicity applies by Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion). Write $\mu=\mathbb{E}[X_1]$ and $S_n=X_1+\cdots+X_n$ for $n\in\mathbb{N}$. Then

$$\frac{S_n}{n}\longrightarrow\mu\quad\text{almost surely}\qquad(n\to\infty),$$

in the sense of Convergence in Probability. The proof rests on the fourth-moment bound $\mathbb{E}\bigl[(S_n-n\mu)^{4}\bigr]\le C n^{2}$ for a constant $C$ depending only on the distribution of $X_1$, together with Markov's and Chebyshev's Inequalities applied to $(S_n-n\mu)^{4}$ and the first Borel–Cantelli lemma.