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 $X_1$ and $X_1^{2}$ have finite \reftext{def:expectation-variance-2026a}{expectation}, and write $\mu=\mathbb{E}[X_1]$. For $n\in\mathbb{N}$ let

Then

in the sense of \ref{def:convergence-modes-2026a}. Explicitly, for every $\varepsilon>0$ and every $n$,

which tends to $0$ as $n\to\infty$; the bound combines \ref{lem:markov-chebyshev-2026a} with the additivity of the variance over independent summands from \ref{lem:expectation-product-independent-2026a}.