# Scheffé’s 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 $X_{n}$ converges to $X$ in distribution (http://planetmath.org/ConvergenceInDistribution): $X_{n}\xrightarrow[]{D}X$.

Title Scheffé’s theorem ScheffesTheorem 2013-03-22 13:14:23 2013-03-22 13:14:23 Koro (127) Koro (127) 6 Koro (127) Theorem msc 60E05