Definition:Convergence in Distribution

Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\sequence {X_n}_{n \mathop \in \N}$ be a sequence of real-valued random variables on $\struct {\Omega, \Sigma, \Pr}$.

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

For each $n \in \N$, let $F_n$ be the cumulative distribution function of $X_n$.

We say that $\sequence {X_n}_{n \mathop \in \N}$ converges in distribution to $X$ if and only if:

$\ds \lim_{n \mathop \to \infty} \map {F_n} x = \map F x$

for all $x$ for which $F$ is continuous.

This is written:

$X_n \xrightarrow d X$