convergence in distribution

A sequence of distribution functions $F_1,F_2,\mathrm{\dots }$ converges weakly to a distributionfunction $F$ if $F_n(t)\to F(t)$ for each point $t$ at which $F$ is continuous.

If the random variables $X,X_1,X_2,\mathrm{\dots }$ have associated distribution functions$F,F_1,F_2,\mathrm{\dots }$, respectively, then we say that $X_n$ converges in distribution to$X$, and denote this by $X_n\stackrel{𝐷}{\to }X$.

This definition holds for joint distribution functions and random vectors as well.

This is probably the weakest of convergence of random variables. Some results involving this of convergenceare the central limit theorems, Helly-Bray theorem, Paul Lévy continuity theorem, Cramér-Wold theorem and Scheffé’s theorem.