- Hi everybody !!!

My question is:

When I have an arbitrary sequence of random variables X(n) , is there a

sequence of constants a(n) such that the quotient X(n)/a(n) converges

almost surely to zero ??????

How can I choose these a(n) ?

Please let me know !!

Thanks a lot !!!

Stoeps - Hi Stoeps,

In the case when (Xn) is a sequence of integrable random variables, I

believe your conjecture is true. If you choose (an) such that:

an>n*(2^n)*E[|Xn|] for all n

Then P(|Xn|/an > 1/n)=P(|Xn|>an/n)>=n*E[|Xn|]/an < 1/2^n

and in particular, the series Sum P(|Xn|/an > 1/n) is finite. Using

borel-cantelli's lemma, with probability one, |Xn|/an <= 1/n

eventually, and in particular, |Xn|/an tends to zero almost surely.

Since the same argument can be applied for other sequences than (1/n)

and (1/2^n), it is possible to choose (an) more optimally than I did.

Regards. Noel. - --- In probability@y..., stoepss@y... wrote:
> Hi everybody !!!

there a

>

> My question is:

>

> When I have an arbitrary sequence of random variables X(n) , is

> sequence of constants a(n) such that the quotient X(n)/a(n)

converges

> almost surely to zero ??????

> How can I choose these a(n) ?

>

> Please let me know !!

>

> Thanks a lot !!!

>

> Stoeps - Hi folks !!

I have here a problem i can't figure out.

Maybe some of you can give me a hint to do this!

Following situation:

Let X(n), where n=0,+1,-1,+2,-2,...., be independent and identically

distributed according to the normal standard distribution Phi.

Then the series of complex-valued random variables

x*X(0) + sum(exp(inx*X(n))/(in)) + sum(exp(-inx*X(-n))/(-in))

(n from 1 to infinity in both sums)

converges almost surely and uniformly in x.

How can I show this and what is the limit ?

Thank you !

Stoeps - Hi again folks !!!

Recently I considered following problem:

For any sequence of random variables X(n) there exists a sequence

of constants a(n) such that the quotient (X(n)/a(n)) converges to 0

almost everywhere.

For integrable random variables Noel told me to pick the a(n)'s

greater than n*(2^n)*E|X(n)|. Then the claim is true using the

Borel Cantelli Lemma. (thanks Noel !!!!)

But how can I choose the a(n) when the random variables are arbitrary

(not necessarily integrable) ?

I think I must use the Borel-Cantelli Lemma again.

Can anybody help me ?

Thank you !!!

Stoeps