Almost Sure Convergence, Convergence in Probability, and Convergence in Distribution

Let $(X_m)_{m\in\mathbb{N}}$ and $X$ be 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 $X$ almost surely if there is an event $A\in\mathcal{F}$ with $P(A)=1$ such that for every $\omega\in A$ the sequence $(X_m(\omega))_{m\in\mathbb{N}}$ converges to $X(\omega)$ in the sense of Limit of a Sequence of Real Numbers.

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

3. $(X_m)$ converges to $X$ in distribution if

for every $t\in\mathbb{R}$ at which the cumulative distribution function $F_X$ is continuous, as a map from the real line $(\mathbb{R},d_{\mathbb{R}})$ into itself.