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Maximum partial sum of sequance of random variables

  1. Jul 27, 2010 #1
    Hi friends/colleagues,

    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.

    Let S be the vector of partial sum of Xs: Si=X1+X2+...+Xi.

    Question: What is the limiting distribution of Maxi(Si), 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 Sadatsafavi.
    Center for Clinical Epidemiology and Evaluation
    University of British Columbia

    mohsen dot safavi at gmail dot com
  2. jcsd
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