Definition:Convergence in Distribution

Definition
Let $\sequence {X_n}_{n \ge 1}$ be a sequence of random variables.

Let $X$ be a random variable.

We say that $\sequence {X_n}$ converges in distribution to $X$ if:


 * $\displaystyle \lim_{n \mathop \to \infty} \map \Pr {X_n \le x} = \map \Pr {X \le x}$

for all $x$ for which the map $x \mapsto \map \Pr {X \le x}$ is continuous.

This is written:


 * $X_n \xrightarrow d X$