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Math Help - maximum partial sum of sequance of random variables

  1. #1
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    maximum partial sum of sequance of random variables

    Hi friends/colleagues,

    I am banging my head to the wall answering a question that seems to be a classic in probability theory (my field is health care so I am far from the field though).

    Let X1, X2, ..., Xn be a sequence of independent, but NOT identically distributed random variables, with E(Xi)=0, and variance of each Xi being UNEQUAL but finite.

    What is the limiting distribution of the maximum partial sum of X? By limiting distribution I mean as n grows to infinity.

    I can also formulate this question slightly differently: is the limiting distribution of partial sum of X a Brownian movement process? In that case the maximum partial sum is maximum distance of Brownian motions from its origin which has a closed formula.

    If this question does not have answer with this assumptions, I need to know what additional assumptions I need to make.

    just in case, one more condition in this problem is that the variance function of X is a 'smooth' function in that If if Xi -> Xj then Var(Xi)->Var(Xj).

    Your help is much appreciated.

    Mohsen
    Last edited by mr fantastic; July 27th 2010 at 07:22 PM.
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  2. #2
    MHF Contributor matheagle's Avatar
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    Kolmogorov's inequality may be of some help....
    Kolmogorov's inequality - Wikipedia, the free encyclopedia
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