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Thursday, January 29, 2015

Record a problem

Dini's Theorem fails when the sequence of functions $\{f_n(x)\}$ is not monotone in $n$, or when the common domain is not a compact set. But in probability theory we still have a similar result that serves as a work-around to obtain uniform convergence.
Problem. Suppose $F_1,F_2,\dots ,F:\R\to [0,1]$ are cumulative distribution functions and that $F_n\to F$ in distribution. Show that \[
\sup_{x\in \R} |F_n(x)-F(x)|\to 0
\] as $n\to \infty$ if $F(x)$ is a continuous function. Give a counte-example when $F(x)$ is not continuous.
Here the common domain $\R$ can be replaced by any kind of interval. We can prove this by contradiction, the only difficulty, if any, is the familiarity with the terms "distribution function" and "convergence in distribution". The former means that it is a nonnegative increasing function that takes value from $0$ to $1$, while the later means that the distributions of a sequence of random variables converge pointwise to the target at the point of continuity of the target distribution (of a target random variable).
張貼者： 於
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