I'm having problems showing this result.
Problem: Let and be simple random variables. Show that with probability 1 if and only if for all there exists an n such that for all m > n.
I haven't made much progress. Hopefully a point in the correct direction will get me going.
I don't understand why you entitled your thread with almost sure convergence ? At first glance, it's not about a.s. convergence !
What do you call "simple rv's" ? And are the rv independent ?
in probability means that
By definition of a limit, we can write :
In particular for , we have
and hm well, maybe you can finish it off... I can't see well at the moment (they're all equalities/equivalences so it solves the "iff" problem)
This problem is 6.1 from Billingsley's Probability and Measure. It is quoted exactly, so no, the random variables are not independent. A simple rv is one with a finite range. And it is, indeed, about a.s. convergence. Billingsley refers to convergence a.s. as convergence with probability 1. It is not a question about convergence in probability.