## almost sure convergence

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• 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
Message 1 of 5 , Mar 10, 2001
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) ?

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:
Message 2 of 5 , Mar 10, 2001
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.
• ... there a ... converges
Message 3 of 5 , Mar 10, 2001
--- In probability@y..., stoepss@y... wrote:
> 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 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
Message 4 of 5 , Mar 22, 2001
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
Message 5 of 5 , Mar 30, 2001
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
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