Chapter 4 exercise BC

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Chapter 4 exercise BC

Postby vkaul1 » Fri Oct 09, 2009 12:04 am

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.
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Re: Chapter 4 exercise BC

Postby mj » Sun Oct 11, 2009 11:31 pm

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.
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Re: Chapter 4 exercise BC

Postby Daniel » Tue Aug 02, 2011 10:25 pm

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.
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