Let $(X_m)_{m\in\mathbb{N}}$ and $X$ be \reftext{def:probability-space-random-variable-2026a}{random variables} on a common probability space $(\Omega,\mathcal{F},P)$ (for convergence in distribution, a common space is not required).

1. $(X_m)$ converges to $X$ \textbf{almost surely} if

with pointwise convergence in the sense of \ref{def:limit-sequence-real-c54-2026a}; the set in question is an event, since it equals $\bigcap_{j}\bigcup_{k}\bigcap_{m\ge k}\{|X_m-X|\le 1/j\}$ over $j,k\in\mathbb{N}$, and $|X_m-X|$ is a random variable by Step 0(a) of the proof of \ref{thm:linearity-monotonicity-integral-2026a}.

2. $(X_m)$ converges to $X$ \textbf{in probability} if for every $\varepsilon>0$,

3. $(X_m)$ converges to $X$ \textbf{in distribution} if

for every $t\in\mathbb{R}$ at which the \reftext{def:distribution-cdf-random-variable-2026a}{cumulative distribution function} $F_X$ is \reftext{def:continuous-at-point-c54-2026b}{continuous}.