convergence in distribution

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A sequence of random variables x1,…, xn,… with corresponding distribution functions F1(x),…Fn(x),… converges in distribution (or weakly) to the random variable x with distribution function F(x) if the sequence of the corresponding distribution functions converges to F at all continuity points of Ffor every ε > 0 starting from some n. The distribution given by F(x) is called the limiting or the asymptotic distribution of xn. This concept allows the approximation of the unknown distribution Fn(x) of an estimator or a test statistic with a known asymptotic distribution F(x). If x = θ is a constant the limiting distribution is degenerate, i.e. collapses to a single point. In this case (but not in general) convergence in distribution also implies convergence in probability.

Subjects: Economics.

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