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CumulantGeneratingFunction—Wolfram Language Documentation

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BUILT-IN SYMBOL

CumulantGeneratingFunction

CumulantGeneratingFunction[dist,t]

gives the cumulant-generating function for the distribution dist as a function of the variable t.

CumulantGeneratingFunction[dist,{t₁,t₂,…}]

gives the cumulant-generating function for the multivariate distribution dist as a function of the variables t₁, t₂, … .

Details

CumulantGeneratingFunction[dist,t] is given by Log[MomentGeneratingFunction[dist,t]].
CumulantGeneratingFunction[dist, {t₁,t₂,…}] is given by Log[MomentGeneratingFunction[dist,{t₁,t₂,…}]].
The i cumulant can be extracted from a cumulant-generating function cgf through SeriesCoefficient[cgf,{t,0,i}]i!.

Examplesopen allclose all Basic Examples (3)

Compute a cumulant-generating function (cgf) for a continuous univariate distribution:

The cgf for a univariate discrete distribution:

The cgf for a multivariate distribution:

Scope (5)

Compute the cgf for a formula distribution:

Find the cgf for a function of random variates:

Compute the cgf for data distribution:

Find the cgf for a truncated distribution:

Find the cgf for the slice distribution of a random process:

Applications (5)

The cumulant-generating function of a difference of two independent random variables is equal to the sum of their cumulant-generating functions with oppositive sign arguments:

Illustrate the central limit theorem:

Find the cumulant-generating function for the standardized random variate:

Find the moment-generating function for the sum of standardized random variates rescaled by :

Find the large limit:

Compare with the moment-generating function of a standard normal distribution:

Find the Esscher premium for insuring against losses following GammaDistribution:

Compare with the definition:

Construct a Bernstein–Chernoff bound for the survival function :

Large approximation of the bound:

Construct Daniel's saddle point approximation to PDF of VarianceGammaDistribution:

Find the saddle point associated with the argument of probability density function :

Select the solution that is valid for all real , including the origin:

The approximation is constructed using the cumulant-generating function at the saddle point:

Find the normalization constant:

Compare the approximation to the exact density:

Properties & Relations (3) Wolfram Research (2010), CumulantGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html. Text Wolfram Research (2010), CumulantGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html.CMS Wolfram Language. 2010. "CumulantGeneratingFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html.APA Wolfram Language. (2010). CumulantGeneratingFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.htmlBibTeX @misc{reference.wolfram_2025_cumulantgeneratingfunction, author="Wolfram Research", title="{CumulantGeneratingFunction}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html}", note=[Accessed: 12-July-2025 ]}BibLaTeX @online{reference.wolfram_2025_cumulantgeneratingfunction, organization={Wolfram Research}, title={CumulantGeneratingFunction}, year={2010}, url={https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html}, note=[Accessed: 12-July-2025 ]}

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