Calculating the distribution of sums of log-gamma random variables

The problem of determining the distribution of sums of (weighted) log-gamma random variables occurs in spectrum sensing for cognitive radio. An infinite series is derived for the characteristic function of log-gamma random variables. Some special cases are identified for which the infinite series can be represented in terms of some well-known functions, greatly improving computational simplicity and robustness. The derived characteristic function representations are used to calculate the PDF of sums of log-gamma random variables. The results are extended to the case of weighted sums of log-gamma random variables. Examples indicate that a Gaussian approximation to a log-gamma sum density can be very inaccurate for smaller numbers of summands. The application of the results to a spectrum sensing problem in cognitive radio is explained.