Maximum partial sum of sequance of random variables

[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 Max_i(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

 

[mmwave]

Dear Mohsen Sadatsafavi,

Thank you for your inquiry regarding the limiting distribution of the maximum partial sum of a sequence of independent, but not identically distributed random variables. This is an interesting question that has been studied extensively in the field of probability theory.

To answer your first question, the limiting distribution of Maxi(Si) is known as the Gumbel distribution. This distribution is a type of extreme value distribution and is commonly used to model the maximum or minimum of a large number of random variables.

To answer your second question, the partial sum of X does not follow a Brownian movement process. Brownian motion is a continuous stochastic process, while the partial sum of X is a discrete process. However, the maximum partial sum can be thought of as the maximum distance from the origin of a random walk, which does have a closed formula.

In order to make further assumptions and find a closed formula for the limiting distribution, you may need to specify the distribution of the individual Xi's. Without additional information, it is difficult to provide a specific answer. However, if the Xi's are normally distributed, then the maximum partial sum will follow a Gumbel distribution.

I hope this information helps. If you have any further questions, please don't hesitate to reach out.