\begin{definition}[Definition:Convergence in Distribution]
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. 
Let $\sequence {X_n}_{n \in \N}$ 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 $X_n$.
Let $F$ be the cumulative distribution function of $X$. 
We say that $\sequence {X_n}_{n \in \N}$ '''converges in distribution''' to $X$ 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: 
:$X_n \xrightarrow d X$
\end{definition}