Scheffé’s theorem

Let $X,X_1,X_2,\mathrm{\dots }$ be continuous random variables in a probability space, whose probability density functions are $f,f_1,f_2,\mathrm{\dots }$, respectively. If $f_n\to f$ almost everywhere (relative to Lebesgue measure,) then $X_n$ converges to $X$ in distribution (http://planetmath.org/ConvergenceInDistribution): $X_n\stackrel{𝐷}{\to }X$.

Title	Scheffé’s theorem
Canonical name	ScheffesTheorem
Date of creation	2013-03-22 13:14:23
Last modified on	2013-03-22 13:14:23
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	6
Author	Koro (127)
Entry type	Theorem
Classification	msc 60E05