The Central Limit Theorem: Use with Caution

The validity of the central limit theorem for the sum of N, K-distributed random phasors is investigated. It is demonstrated that the number of phasors that must be summed to obtain an amplitude distribution that can be approximated by a Rayleigh depends strongly on the underlying K-distribution, and is of order 200 for Weibull statistics. It is quite often stated that when N independent, random phasors are summed (e.g., N clutter cells) the amplitude of the sum is approximately Rayleigh (i.e., in-phase and quadrature components are each gaussian) if N is sufficiently large. The question we examine here is how large is "sufficiently large"? In order to answer this, consider the normalized sum of N independent, random phasors.