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## Homework Statement

Let a probability space be [itex](Ω, \epsilon, P)[/itex]. A set of random variables X

_{1},...,X

_{n}

Give an example where [itex]I_{p}(lim inf_{n -> ∞}X_{n}[/itex]) < [itex]lim inf_{n -> ∞}I_{p}(X_{n})[/itex]

**The attempt at a solution**

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

and [itex]lim inf_{n -> ∞}I_{p}(X_{n})[/itex] = [itex]lim inf_{n -> ∞}E[X_{n}][/itex]

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.