The distribution of second order moment statistics in a normal system

Abstract

Let x be a normally distributed variable, which may without loss of generality be measured from the mean of the distribution, so that E(x) = 0, E denoting mathematical expectation. Then x satisfies the differential relationwhere k2 (otherwise σ2 or (2h2)−1) is the semi-invariant of order 2. Also k1 = 0, and we know that kr = 0 for r > 2.