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almost sure convergence

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  • stoepss@yahoo.de
    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
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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 X(n)/a(n) converges
      almost surely to zero ??????
      How can I choose these a(n) ?

      Please let me know !!

      Thanks a lot !!!

      Stoeps
    • vaillant@probability.net
      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
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        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.
      • vaillant@probability.net
        ... there a ... converges
        Message 3 of 5 , Mar 10, 2001
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          --- 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
        • stoepss@yahoo.de
          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
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            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
          • stoepss@yahoo.de
            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
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              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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