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With this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given probability distribution.
Convergence in distribution is the weakest form of convergence typically discussed, since it is implied by all other types of convergence mentioned in this article. However, convergence in distribution is very frequently used in practice; most often it arises from application of the central limit theorem.
A sequence X1, X2,... of real-valued random variables is said to converge in distribution, or converge weakly, or converge in law to a random variable X if
for every number x ∈ R {\displaystyle x\in \mathbb {R} } at which F is continuous. Here, Fn and F are the cumulative distribution functions of the random variables Xn and X, respectively.