@James:Thanks, you are right, probability of exceeding T/2 might be very close to 1/2. I was not very clear about the bound I want to prove for the sampled sum. I want to show that sum cannot be much smaller than T/2, say $T/2-\epsilon$ with high probability. The problem I am facing is that since $X_i$s can be large, $0\leq X_i\leq T$ for all $i$, bounded differences inequalities do not seem to give much. Suggestions or pointers are welcome.