Discrete random bounds for general random variables and applications to reliability

We here propose some new algorithms to compute bounds for (1) cumulative density functions of sums of i.i.d. nonnegative random variables, (2) renewal functions and (3) cumulative density functions of geometric sums of i.i.d. nonnegative random variables. The idea is very basic and consists in bounding any general nonnegative random variable X   by two discrete random variables with range in hNhN, which both converge to X as h goes to 0. Numerical experiments are lead on and the results given by the different algorithms are compared to theoretical results in case of i.i.d. exponentially distributed random variables and to other numerical methods in other cases.