Let $(X_m)_{m\in\mathbb{N}}$ and $X$ be \reftext{def:probability-space-random-variable-2026a}{random variables}, not necessarily on a common probability space, and let $\mathbb{R}$ denote the \reftext{def:real-numbers-c54-2026c}{real numbers}. Call a function $f:\mathbb{R}\to\mathbb{R}$ an \textbf{admissible test function} if $f$ is bounded, $f$ is a \reftext{def:ck-map-euclidean-open-set-2026b}{$C^3$ map} on $\mathbb{R}=\mathbb{R}^1$, and its first, second, and third derivatives are bounded. Then the following hold.

1. For every admissible test function $f$ and every random variable $Y$, the composition $f\circ Y$ is a random variable with finite \reftext{def:expectation-variance-2026a}{expectation}; in particular $\mathbb{E}[f(X_m)]$ and $\mathbb{E}[f(X)]$ are defined real numbers.

2. If

for every admissible test function $f$, then $X_m\to X$ in distribution, in the sense of \ref{def:convergence-modes-2026a}.