Alternative Descriptions for Random Variables

Project Type: Master/Diploma Thesis
Student: Christian Knoll

Short Description

The cumulants of a probability distribution are the coefficients of a Taylor series expansion of the logarithm of the characteristic function. The more cumulants of such a probability distribution are known, the better will therefore be the match between the true and the approximated distribution function.

While a Taylor series expansion is possible for both aperiodic and periodic characteristic functions (corresponding to continuous and discrete random variables), a Fourier series expansion is only possible in the periodic case. Also in this case it can be assumed that the more Fourier coefficients of this expansion are known, the better will be the approximation.

This project/thesis aims at analyzing the benefits and drawbacks of a Fourier series compared to a Taylor series, especially when it comes to an incomplete set of coefficients (truncation of the series). Since the Fourier series of the logarithm of the characteristic function is the cepstrum of the associated probability mass function, cepstrum theory [2] will give an insight to which of the coefficients are important, and to the effect of series truncation. Furthermore, since quantization of a random variable leads to a periodic extension of the characteristic function [1], the mathematical relationship between the cumulants of the continuous and the Fourier coefficients of the discrete random variable is of special interest.

Results

The thesis has been completed and the results are currently in preparation for publication. For more information please contact the author or the supervisor.

References

[1] Widrow, B.; Kollar, I; Ming-Chang Liu: "Statistical theory of quantization", IEEE Transactions on Instrumentation and Measurement, 1996

[2] Oppenheim, A.V.; Schafer, R.W.: "From frequency to quefrency: A history of the cepstrum", IEEE Signal Processing Magazine, 2004