[Math] Probability Theory, Chernoff Bounds, Sum of Independent (but not identically distributed) r.v

Question

Is there any known bound on sum of independent but not identically distributed geometric random variables?
I have to show that the tail of the sum drops exponentially (like in the Chernoff bounds for the sum of iid geom. variables).

Formally, if $X_i \sim Geom(p_i)$, $X = \sum_{i=1}^n X_i$, and $E[X]=\Theta(n)$,

Is it possible to show that $\Pr(X < 2E[X]) > 1 – \delta ^n$, where $\delta < 1$?

Thank you in advance,
Michael.

Answer 1

Hoeffding's inequality can be used but the random variables need to be bounded to get meaningful bounds. http://en.wikipedia.org/wiki/Hoeffding%27s_inequality.