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Cumulant-Generating Function

Let M(h) be the moment-generating function, then the cumulant generating function is given by

K(h)=lnM(h)
(1)
=kappa_1h+1/(2!)h^2kappa_2+1/(3!)h^3kappa_3+...,
(2)

where kappa_1, kappa_2, ..., are the cumulants.

If

L=sum_(j=1)^Nc_jx_j
(3)

is a function of N independent variables, then the cumulant-generating function for L is given by

K(h)=sum_(j=1)^NK_j(c_jh).
(4)

See also

Cumulant, Moment-Generating Function

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972.Kenney, J. F. and Keeping, E. S. "Cumulants and the Cumulant-Generating Function" and "Additive Property of Cumulants." §4.10-4.11 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 77-80, 1951.

Referenced on Wolfram|Alpha

Cumulant-Generating Function

Cite this as:

Weisstein, Eric W. "Cumulant-Generating Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cumulant-GeneratingFunction.html

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