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.