# Maximum partial sum of sequance of random variables

1. Jul 27, 2010

### shoomchool

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).