# Thread: Almost Sure Convergence Criterion

1. ## Almost Sure Convergence Criterion

Hello all,

I'm having problems showing this result.

Problem: Let $\displaystyle Z_1, Z_2, ...$ and $\displaystyle Z$ be simple random variables. Show that $\displaystyle Z_n \to Z$ with probability 1 if and only if for all $\displaystyle \epsilon > 0$ there exists an n such that $\displaystyle P(|Z_k - Z| < \epsilon, n \le k \le m) > 1 - \epsilon$ for all m > n.

I haven't made much progress. Hopefully a point in the correct direction will get me going.

2. Originally Posted by Guy
Hello all,

I'm having problems showing this result.

Problem: Let $\displaystyle Z_1, Z_2, ...$ and $\displaystyle Z$ be simple random variables. Show that $\displaystyle Z_n \to Z$ with probability 1 if and only if for all $\displaystyle \epsilon > 0$ there exists an n such that $\displaystyle P(|Z_k - Z| < \epsion, n \le k \le m) > 1 - \epsilon$ for all m > n.

I haven't made much progress. Hopefully a point in the correct direction will get me going.
I think you missed something out in the probability (|Z_k-Z|< ????) but it seems like you will need Borel-Cantelli at some point.

3. Yes, sorry, it is supposed to be $\displaystyle P(|Z_k - Z| < \epsilon, n \le k \le m) > 1 - \epsilon$ (which I've now edited so that it is correct). And yeah, I assumed I would need Borel-Cantelli but that is more of a finishing move to one of the directions whereas I can't even get started.

4. Hello,

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 ?

$\displaystyle Z_n\to Z$ in probability means that $\displaystyle \forall \epsilon>0,\lim_{n\to\infty} P(|Z_m-Z|<\epsilon)=1$
By definition of a limit, we can write :
$\displaystyle \forall \epsilon>0,\forall \tilde{\epsilon}>0,\exists n\in\mathbb{N},\forall m>n,P(|Z_m-Z|<\epsilon)>1-\tilde{\epsilon}$

In particular for $\displaystyle \tilde{\epsilon}=\epsilon$, we have $\displaystyle \forall \epsilon>0, \exists n\in\mathbb{N},\forall m>n, P(|Z_m-Z|<\epsilon)>1-\epsilon$

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)

5. 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.

6. Originally Posted by Guy
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.
Yeah sorry, I misread the "with probability 1".
I'll definitely go to bed