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convergence in distribution (Definition)

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

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

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

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



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See Also: weak convergence

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Cross-references: Scheffé's theorem, Cramér-Wold theorem, Paul Lévy continuity theorem, Helly-Bray theorem, central limit theorems, random vectors, joint distribution functions, random variables, continuous, point, converges, distribution functions, sequence
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This is version 8 of convergence in distribution, born on 2002-12-10, modified 2005-02-11.
Object id is 3708, canonical name is ConvergenceInDistribution.
Accessed 6203 times total.

Classification:
AMS MSC60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)

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