Expected values in Probability space

[Lily@pie]

Homework Statement

Let a probability space be $(Ω, \epsilon, P)$. A set of random variables X₁,...,X_n

Give an example where $I_{p}(lim inf_{n -> ∞}X_{n}$) < $lim inf_{n -> ∞}I_{p}(X_{n})$


The attempt at a solution

I know that $I_{p}(lim inf_{n -> ∞}X_{n}$)=$E[lim inf_{n -> ∞}X_{n}$]
and $lim inf_{n -> ∞}I_{p}(X_{n})$ = $lim inf_{n -> ∞}E[X_{n}]$

I think I need to find a sequence of X_n such that lim inf X_n will have a smaller value than all the individual expected value, E[X_n].

Am I on the correct path? I'm kind of stuck here and not sure how to proceed.

Would be really really thankful for the help.

 

[mfb]

How is $\displaystyle \liminf_{n \to \infty} X_n$ defined? The limit of the "smallest value" of those random variables?
In that case, I don't see how both can be equal apart from trivial X_i.

 

[micromass]

Equality actually holds for surprisingly many cases. See for example, the dominated convergence theorem. So you'll need to look at sequences of random variables which fail that theorem.
Without giving away too much, try to make a nonconvergent sequence $X_n$ such that $E[X_n]$ are all constant.

 

[mfb]

Okay, then I don't understand how to evaluate $\displaystyle \liminf_{n \to \infty} X_n$.

 

[micromass]

If $X = \liminf_n X_n$, and $\omega$ is an outcome, then it is defined by
$$X(\omega) = \liminf_n X_n(\omega)$$

So it's just the pointswise inferior limit.

 

[Lily@pie]

try to make a nonconvergent sequence $X_n$ such that $E[X_n]$ are all constant.

I saw this example...

Ω=N
P(ω)=2^-ω, ω in Ω
X_n(ω)=2ⁿδ_ω,n

I can see that X_n is not convergent.

But I'm not quite sure in computing the integrals... I'm very bad with the lim inf concept >_< So, I have written my attempt on evaluating the integrals and reasoning that I've used.

My attempt,
lim inf X_n
=lim inf [2ⁿδ_ω,n]
Since this takes in values 0 or 2ⁿ
=0

E[X_n]
=I_P[2ⁿδ_ω,n]
= ∫2ⁿδ_ω,n2^-ωdω
When ω=n, this reduces to ∫δ_ω,ndω
=δ_ω,n

I'm not very sure about the way I evaluate the integral. Would be very helpful for some guidance.