Processing math: 100%

Sunday, December 7, 2014

Exam Problem from Probability Class

These two days PG students in our office struggled for their take-home final exam with dead-line two days after it was released. They were stuck with the following problem (and so I was, pleasantly, asked for help):
Problem. Let (X,μ) be a probability measure space such that fnμ0 and that Var(fn)=1, prove that E(fn)0.

Here Var(fn)=Xf2ndμ(Xfndμ)2 and E(fn)=Xfndμ.

In the sequel we denote f=Xfdμ and Af=Afdμ. Then E(fn) is also written as fn.

In the proof below we need an elementary inequality from probability that easily follows from Chebyshev's inequality: μ{xX:|fn(x)E(fn)|a}Var(fn)a2.
Proof.
Suppose on the contrary that E(fn)0, then there is an ϵ0>0 such that |fnk|ϵ0. It follows that ϵ0|{|fnk|1}fnk|+|{|fnk|<1}fnk|{|fnk|1}f2nkμ{|fnk|1}+|fnk|χ{|fnk|<1}f2nkμ{|fnk|1}+|fnk|χ{|fnk|<1}. Since fnk0 in measure, there is a subsequence fnkp that converges to 0 pointwise μ-a.e.. By Lebesgue Dominated Convergence Theorem, |fnkp|χ{|fnkp|<1}0. So the above becomes ϵ0¯limpf2nkpμ{|fnkp|1}. We will finish the proof by showing fn is in fact bounded in n, and so is f2n since 1=Var(fn)=f2n(fn)2, and then the above inequality yields a contradiction that ϵ00.

Now we turn to the boundedness of  {fn}, the only tricky part of this problem. For this, observe that for any measurable f,g on X, μ{|f|2ϵ}μ{|fg|+|g|2ϵ}μ{|fg|ϵ}+μ{|g|ϵ}. In particular, let's take f to be the constant function E(fn), and g=fn, then μ{xX:|E(fn)|2ϵ}μ{|E(fn)fn|ϵ}+μ{|fn|ϵ}1ϵ2+μ{|fn|ϵ}, the last inequality follows from μ{|fnE(fn)|ϵ}Var(fn)ϵ2. Taking ¯lim on both sides above, we get ¯limμ{xX:|E(fn)|2ϵ}1ϵ2 by convergence in measure. Take ϵ=2, there is an N such that for every n>N, μ{xX:|E(fn)|22}<1, and therefore {xX:|E(fn)|4}=, and thus |E(fn)|<4 for n>N.

Remark. After examining the proof the condition that Var(fn)=1 can be replaced by that the sequence {Var(fn)}n=1 is bounded, since we can generalize (???) to
¯limμ{xX:|E(fn)|2ϵ}Var(fn)ϵ2.

No comments:

Post a Comment