Convergence in distribution

Suppose that F1F2, … is a sequence of cumulative distribution functions corresponding to random variables X1X2, …, and that F is a distribution function corresponding to a random variable X. We say that the sequence Xn converges towards X in distribution, if

$\lim_{n\rightarrow\infty} F_n(a) = F(a),$

for every real number a at which F is continuous. Since F(a) = Pr(X ≤ a), this means that the probability that the value of X is in a given range is very similar to the probability that the value of Xn is in that range, provided n is sufficiently large. Convergence in distribution is often denoted by adding the letter $\mathcal D$ over an arrow indicating convergence:

$X_n \, \xrightarrow{\mathcal D} \, X$

Small d is also possible, although less common.

Convergence in distribution is the weakest form of convergence, and is sometimes called weak convergence (main article: weak convergence of measures). It does not, in general, imply any other mode of convergence. However, convergence in distribution is implied by all other modes of convergence mentioned in this article, and hence, it is the most common and often the most useful form of convergence of random variables. It is the notion of convergence used in the central limit theorem and the (weak) law of large numbers.

A useful result, which may be employed in conjunction with law of large numbers and the central limit theorem, is that if a function  gR → R  is continuous, then if  Xn  converges in distribution to  X, then so too does  g(Xn)  converge in distribution to  g(X). (This may be proved using Skorokhod’s representation theorem.) This fact could be taken as a definition for the convergence in distribution.

Convergence in distribution is also called convergence in law, since the word “law” is sometimes used as a synonym of “probability distribution.”