convergence in distribution

For i.i.d random variables, we can always say \(X_n\overset{d}{\rightarrow}X\) since \[\begin{align}%\label{eq:union-bound} F_{X_n}(x)=F_X(x), \qquad \textrm{ for all }x. \end{align}\]

\[\therefore\]

\[\begin{align}%\label{eq:union-bound} \lim_{n \rightarrow \infty} F_{X_n}(x)=F_X(x), \qquad \textrm{ for all }x. \end{align}\]