Let $(X_m)_{m\in\mathbb{N}}$ be a \reftext{def:independence-events-rvs-2026a}{sequence of independent and identically distributed random variables} on a probability space $(\Omega,\mathcal{F},P)$ such that $X_1^{4}$ has finite \reftext{def:expectation-variance-2026a}{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 \ref{thm:linearity-monotonicity-integral-2026a}). Write $\mu=\mathbb{E}[X_1]$ and $S_n=X_1+\cdots+X_n$ for $n\in\mathbb{N}$. Then

in the sense of \ref{def:convergence-modes-2026a}. 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 \ref{lem:markov-chebyshev-2026a} applied to $(S_n-n\mu)^{4}$ and the first \reftext{lem:borel-cantelli-2026a}{Borel–Cantelli lemma}.