Cumulant-Generating Functions for Normal and Lognormal Variables

Summary

The document defines the cumulant-generating function as the logarithm of the moment-generating function and relates its series coefficients to the distribution’s cumulants. It also states that for a weighted sum of independent variables, the cumulant-generating function is the sum of each variable’s cumulant-generating function evaluated at the correspondingly scaled argument. This provides a compact way to derive cumulants for sums without first constructing the full distribution.

For a normal variable, the moment-generating function leads to a cumulant-generating function with a linear term for the mean and a quadratic term for the variance; higher-order cumulants vanish. The answer notes that this expression can be evaluated directly in Matlab. It also warns that the lognormal distribution does not have a moment-generating function as defined here, so this method cannot be applied to it in the same way. The explanation is brief and gives no Matlab implementation or worked calculation for a weighted sum.

Key ideas

• The cumulant-generating function is the logarithm of the moment-generating function.
• Its series expansion encodes cumulants as coefficients of successive powers of the argument.
• For independent variables, the cumulant-generating function of a weighted sum is the sum of the scaled component functions.
• For a normal distribution, the function contains terms for the mean and variance.
• The lognormal distribution has no moment-generating function, limiting this approach for that distribution.

Tags

# How can I calculate the Cumulant-Generating Function in Matlab?


# How can I calculate the Cumulant-Generating Function in Matlab?












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

$$K(h)=\text{ln}M(h)=\\ =\kappa_1h+\frac{1}{2!}h^2\kappa_2+\frac{1}{3!}h^3\kappa_3+\ldots$$ where $\kappa_1, \kappa_2, \ldots$, are the cumulants.

If $L=\sum_{j=1}^Nc_jx_j$ is a function of $N$ independent variables, then the cumulant-generating function for $L$ is given by $$ K(h)=\sum_{j=1}^NK_j(c_jh). $$

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/9913

I am trying to make things clear with this answer. In the case of the normal distribution it holds that the moment generating function (mgf) is given by $$ M(h) = \exp(\mu h + \frac12 \sigma^2 h^2), $$ where $\mu$ is the mean and $\sigma^2$ is the variance. Thus the cumulant generating function $C(h)$ which is given by $C(h) = \ln (M(h))$ reduces to $$ C(h) = \mu h + \frac12 \sigma^2 h^2. $$ I am sure you can evaluate this in Matlab.

In the case of log-normal the mgf is not defined.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.