The fast Fourier transform and its applications
The fast Fourier transform and its applications
Numerical recipes in C (2nd ed.): the art of scientific computing
Numerical recipes in C (2nd ed.): the art of scientific computing
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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. In this paper, we show how this problem can be easily overcome in actuarial applications 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 ε =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 show that, generally speaking, the approximated method performs better (also thanks to the substantial simplification obtained by summing, after each convolution, the probabilities corresponding to the same numerical realizations) than both the exact methods.