Sums of Independent (but not identically distributed) Random Variables

[sv79]

I am looking for a Hoeffding-type result that bounds the tail of a sum of independent, but not identically distributed random variables. Let X_1,..,X_n be independent exponential random variables with rates k_1,...,k_n. (Note: X_i's are unbounded unlike the case considered by Hoeffding)

Let S= \sum_{i=1}^{n} X_i. I am interested in bounding P(S>a). I am looking for tighter bounds than Markov's Inequality and Chebyshev's Inequality. Is anyone here aware of well-known results in this direction?

 

[bpet]

independent exponential random variables

Are the parameters all distinct? (this would simplify the analysis considerably)

 

[sv79]

Bpet:
Yes, the parameters are all distinct. Perhaps, you are thinking of a Hypoexponential distribution (also called a Generalized Erlang distribution, I think), which in my case it certainly is. The question is can we get a clean (easily usable like the Hoeffding, Chernoff bounds etc.) tail inequality for the sum?

 

[bpet]

I don't know if there are any simple inequalities (simpler than the matrix exponential formula) but maybe some of the Bernstein inequalities (tricky bit is working out the central moment growth rate). Also from memory there are some tail prob inequalities involving an integral of the characteristic function over some small neighbourhood of zero. Sorry I couldn't be of more help but I'm keen to hear how it goes.