PlanetMath: Scheffé's theorem

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Scheffé's theorem

(Theorem)

Let $X,X_1,X_2,\dots$ be continuous random variables in a probability space, whose probability density functions are $f,f_1,f_2,\dots$ , respectively. If $f_n\rightarrow f$ almost everywhere (relative to Lebesgue measure,) then converges to in distribution: $X_n\xrightarrow[]{D} X$ .

"Scheffé's theorem" is owned by Koro.

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Cross-references: converges, Lebesgue measure, almost everywhere, probability density functions, probability space, continuous random variables
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This is version 3 of Scheffé's theorem, born on 2002-12-10, modified 2008-05-01.
Object id is 3712, canonical name is ScheffesTheorem.
Accessed 2441 times total.

Classification:

60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)

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