On the Sums of Independently Distributed Pareto Variates

An extension and generalization of the technique of RM-5198, in which the Pareto probability density distribution, which occurs in economic studies of the distributions of incomes, is shown to apply to the problem of detection of radar signals of unknown frequency. This study shows that the asymptotic results derived in RM-5198 on the distribution of the sum of N Pareto variates, noted as X, are valid over a much broader range than originally indicated. The distribution of X (the sum of all the individual radar pulse returns in a resolution cell) is characterized by a parameter alpha, which is related to the signal-to-noise ratio when this model is applied to detection theory. The limiting distribution of X (normalized) as N approaches infinity is obtained when alpha is between 0 and 1, and 1 and 2. In the former case, the limiting distribution is shown to belong to the class of stable probability densities concentrated on the positive axis. In the latter case, the limiting distribution of the mean of X is shown to be a probability density function with unit mass at the mean of X and zero mass elsewhere.