A digression on cumulant expansions

The identities (7.63) and (7.64) are very useful because exponential functions of random variables of the forms that appear on the left sides of these identities are frequently encountered in practical applications. For example, we have seen (cf. Eq. (1.5)) that the average (e ), regarded as a function of a, is a generating function for the moments of the random variable z (see also Section 7.5.4 for a physical example). In this respect it is useful to consider extensions of (7.63) and (7.64) to non-Gaussian random variables and stochastic processes. Indeed, the identity (compare Problem 7.8) 

Problem 7.9. Use the procedure described in Appendix 7C to express the fourth cumulant in tenns of the moments (z ) rt = 1,2,3,4. Show that the third and fourth cumulants of the Gaussian distribution function P(z) = y/ a/Ti) exp[—az ] 7 = —oo. oo vanish. 

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