Homework Help: Independant RV's and the Borel-Cantelli Lemmas

1. Jan 15, 2012

Firepanda

The first part I can do no problem!

The second part... well I've really no idea what to do with that S term, I assume it has something to do with the strong law of large numbers? (w is an element in the set of outcomes)

Borel-Cantelli Lemmas

Law of Large Numbers

Does anyone know where to start?

Thanks

edit:

I have an exmple of strong convergence showing that

if P(w : lim Xn(w) = X(w) )

then this is identically equal to

P(Xn -> X) = 1 , almost surely

So I guess I just have to show that P(Xn -> X) = 1,

So show: P(Sn(w)/n -> -1) = 1?

Probably going the wrong way around it.. not sure how Borel Cantelli links to it.

Last edited: Jan 15, 2012
2. Jan 15, 2012

Ray Vickson

It is probably simpler to re-write the problem a bit: let $X_n = -1 + Y_n,$ where $$\Pr\{Y_n = 0 \} = 1 - \frac{1}{n^2}, \mbox{ and } \Pr \{Y_n = n^2 \} = \frac{1}{n^2}.$$ Essentially, you want to show that $\sum_{k=1}^n Y_k / n \rightarrow 0$ w.p. 1. If $A = \{ Y_n > 0 \mbox{ infinitely often } \},$ can you show that Pr{A} = 0?

RGV

3. Jan 15, 2012

Firepanda

Hi RGV, you always answer my questions it seems! :)

So I need to show given the event En = Yn>0

that the sum from n=1 to inf of P(En) is finite (Borel Cantelli)

So this is the sum of n=1 to inf of 1/n2?

Which is pi^2/6, and so the probability is 0!

Correct?

4. Jan 15, 2012

Firepanda

Also any idea on the second part of this question?

It looks like the same structure..

Thanks!

5. Jan 16, 2012

bump..!