Algorithms for the Sum of Discrete Random Variables. Actuarial Applications

In literature, the sum of discrete random variables becomes a problem of heavy (and often impracticable) computation no sooner does the number of convolutions exceed few units (at most in actuarial applications). In this paper, we show how this problem can be easily overcome when using random variables with integer (positive, negative, or null) or referable to integer numerical realizations but not necessarily identically distributed. Under the above-mentioned condition, we illustrate in particular two exact methods and an approximated one for calculating convolution: - the first exact method is based on the well-known Fast Fourier Transform (FFT); - the second exact method is derived from the classical approach using Discrete Fourier Transform (DFT) by means of algebraic manipulations; - the third method is derived from the definition of convolution and it is approximated by neglecting the probabilities less than a given bound ? =10-h (51?h?100)**. As for the error bounds of the approximated method, it is worth noting that the results obtained by this method differ in relative terms from the corresponding exact values of less than 10-9. This can be tested by comparing the convoluted probability distribution obtained by the approximated method with the one obtained by the other two methods and by also comparing the first four moments with those computed directly on the original random variables. The results (in particular the exact and the approximated probability distribution) are identical in practice. It does not exist therefore the problem of a difference along the tail. As a consequence, although the proposed method is an “approximated method” under a mathematical point of view, it can be considered an “exact method” in the actuarial applications. As for the efficiency of calculation, we have to distinguish between the simple sum of discrete random variables and the calculation of compound distributions with prefixed counting distributions (i.g. Poisson, Negative Binomial, Pareto): - in the first case, the approximated method and the second exact method are similar but the approximated method gives further information about the random variables (for instance, the information about the independence using some properties of the characteristic functions); - in the second case, only the approximated method is applicable in practice. Finally, in the conditions of interest, the exact method using FFT is less efficient than the other methods and it has a more limited application field.