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 suppose $\sigma^{2}=\operatorname{Var}(X_1)>0$; write $\mu=\mathbb{E}[X_1]$, $\sigma$ for the positive square root of $\sigma^{2}$ (\ref{thm:nonnegative-real-has-unique-square-root-2026a}), and $S_n=X_1+\cdots+X_n$. Then

where $Z$ is a \reftext{def:standard-normal-distribution-2026a}{standard normal} random variable; explicitly, in the sense of \ref{def:convergence-modes-2026a},

the convergence holding at every $t$ because $\Phi$ is continuous everywhere by claim 3 of \ref{thm:gaussian-integral-2026a}.