Probability distribution of the sum of N complex random variables

A method is described for the investigation of the statistics of the sum of N complex random variables. The method allows one to obtain the exact mean value of the squared modulus of the sum and its moments for any value of N. The method is applied to a problem of interest in laser propagation through the atmospheric turbulence. In this case the variables are log-normally distributed in amplitude and normally distributed in phase. The effect of the diameter of the receiving aperture is obtained.