## Chapter 4 exercise BC

This forum is to discuss details in the book "Probability with Martingales" by David Williams.

### Chapter 4 exercise BC

The exercise on Page 40 of the book is confusing even after the hint. It says that if S=/sum(pn)</infty then prod(1-pn)>0. Can't we say that each term is >0 so the product needs to be positive or he is worried that the product goes to zero.
I don't know how to extend the S<1 case for which prod(1-pn)>1-S to the general case.
vkaul1

Posts: 20
Joined: Thu Aug 06, 2009 10:19 pm

### Re: Chapter 4 exercise BC

The issue is definitely to show that the limit is positive rather than zero.

If you can prove it for S < 1< i am guessing that you can use a scaling argument but will have to think about it some more.
mj

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

### Re: Chapter 4 exercise BC

I know this topic is pretty old now but i've been using this book as self study before my final year as an undergrad so just thought I would try and outline the approach I was going to try and would like to know if you felt I was heading down the right track..

If S:= Sum{p_n} < infinity, then there is N s.t. S_N := Sum_{n>N}(p_n) -> 0. Then the Prod_{n>N}(1-p_n) >= 1 - S_ N and taking limits would suggest Prod_{n>N}(1-p_n) -> 1.

Then Prod(1-p_n) = Prod_{1<=n<=N}(1-p_n) * Prod_{n>N}(1-p_n) and using the result above suggests this tends towards Prod_{1<=n<=N} (1-p_n). And since the product is now finite and each (1-p_n) > 0 we have Prod(1-p_n) > 0.
Daniel

Posts: 1
Joined: Tue Aug 02, 2011 10:04 pm