Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.7 GENERATING FUNCTIONS

Comparing this expression with (30.92), we find thatκ 1 =μ,κ 2 =σ^2 and all other
cumulants are equal to zero.

We may obtain expressions for the cumulants of a distribution in terms of its

moments by differentiating (30.92) with respect totto give

dKX

dt

=

1

MX

dMX

dt

.

Expanding each term as power series intand cross-multiplying, we obtain

(

κ 1 +κ 2 t+κ 3

t^2

2!

+···

)(

1+μ 1 t+μ 2

t^2

2!

+···

)

=

(

μ 1 +μ 2 t+μ 3

t^2

2!

+···

)

,

and, on equating coefficients of like powers ofton each side, we find

μ 1 =κ 1 ,

μ 2 =κ 2 +κ 1 μ 1 ,

μ 3 =κ 3 +2κ 2 μ 1 +κ 1 μ 2 ,

μ 4 =κ 4 +3κ 3 μ 1 +3κ 2 μ 2 +κ 1 μ 3 ,

..

.

μk=κk+k−^1 C 1 κk− 1 μ 1 +···+k−^1 Crκk−rμr+···+κ 1 μk− 1.

Solving these equations for theκk, we obtain (for the first four cumulants)

κ 1 =μ 1 ,

κ 2 =μ 2 −μ^21 =ν 2 ,

κ 3 =μ 3 − 3 μ 2 μ 1 +2μ^31 =ν 3 ,

κ 4 =μ 4 − 4 μ 3 μ 1 +12μ 2 μ^21 − 3 μ^22 − 6 μ^41 =ν 4 − 3 ν^22. (30.93)

Higher-order cumulants may be calculated in the same way but become increas-

ingly lengthy to write out in full.

The principal property of cumulants is their additivity, which may be proved

by combining (30.92) with (30.90). IfX 1 ,X 2 , ...,XNare independent random

variables andKXi(t)fori=1, 2 ,...,Nis the CGF forXithen the CGF of

SN=c 1 X 1 +c 2 X 2 +···+cNXN(where theciare constants) is given by

KSN(t)=

∑N

i=1

KXi(cit).

Cumulants also have the useful property that, under a change of originX→

X+athe first cumulant undergoes the changeκ 1 →κ 1 +abut all higher-order

cumulants remain unchanged. Under a change of scaleX→bX, cumulantκr

undergoes the changeκr→brκr.