Distribution of Sum of Identically Distributed Exponentially Correlated Gamma-Variables

Abstract

A distribution of a sum of identically distributed Gamma-variables correlated according to an "exponential" autocorrelation law $\rho_{kj} = \rho^{|k - j|}(k, j = 1, \cdots n)$ where $\rho_{kj}$ is the correlation coefficient between the $k$th and $j$th random variables and $0 < \rho < 1$ is a given number is derived. An "approximate" distribution of the sum of these variables under the assumption that the sum itself is a Gamma-variable is given. A comparison between exact and approximate distributions for certain values of the correlation coefficient, the number of variables in the sum and the values of parameters of the initial distributions is presented.