Using the Meixner Characteristic Function to Find Cumulants

Summary

The document introduces the Meixner distribution through its characteristic function and asks how to obtain its first, second, and fourth cumulants. It defines cumulants as derivatives of the logarithm of the characteristic function, evaluated at zero, with powers of the imaginary unit accounting for the order of differentiation.

This sets up a symbolic or analytic calculation, but the document contains no derivation or cumulant values. It therefore illustrates how a distribution’s characteristic function can be used to define cumulants, while leaving the requested computation unanswered. No parameter assumptions, software procedure, or checks on the resulting expressions are provided.

Key ideas

• Cumulants can be defined through derivatives of the log characteristic function at zero.
• The Meixner characteristic function is specified in terms of its distribution parameters.
• The document asks for the first, second, and fourth cumulants but does not provide their values or a derivation.

Tags

# Cumulants of Meixner distribution


# Cumulants of Meixner distribution












Hey characteristic function of Meixner distribution is: $$\Phi(u)=\left(\frac{\cos(\beta/2)}{\cosh((\alpha u-i\beta)/2}\right)^{2\delta}$$ I need to calculate the first, second, and fourth cumulant of this distribution i.e $c_n=\frac{1}{i^n}\frac{\partial^n}{\partial u^n}\ln(\Phi(u))$ at $u=0$. Does anyone have the appropriate software and could give me these values?

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