almost sure convergence
- 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 !!!
- 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.
- --- 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 !!!
- Hi folks !!
I have here a problem i can't figure out.
Maybe some of you can give me a hint to do this!
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 !
- 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
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 !!!