Communications in Information and Systems

Volume 20 (2020)

Number 2

Mathematical Engineering: A special issue at the occasion of the 85th birthday of Prof. Thomas Kailath

Guest Editors: Ali H. Sayed, Helmut Bölcskei, Patrick Dewilde, Vwani Roychowdhury, and Stephen Shing-Toung Yau

Stochastic models based on moment matching

Pages: 209 – 248



Patrick Dewilde (Systems of Signal Processing Department, Faculty of Electronics, Wroclaw University of Technology, Wroclaw, Poland; and Institute of Advanced Study, Technische Universität München, Garching, Germany)


The paper considers interpolating models for non-linear, non-Gauss stochastic variables and processes, given a well-ordered set of moments of increasing order. The proposed models use a characterization with independent parameters, much in the style of the Schur–Levinson parametrization for the linear, Gaussian case (a topic to which Tom Kailath made seminal contributions), but very different from it, given the different kind of structured matrices involved (Hankel-like instead of Toeplitz). The paper starts out with a review of the classical Hamburger–Akhiezer–Jacobi parametrization for one stochastic variable, using a (non-classical) dynamical system theory approach. Next, the paper generalizes these results to the multivariable case, and presents a detailed generalized Jacobi-like (independent) parametrization for two variables. Like in the Schur–Levinson case, such parametrizations succeed in characterizing models that interpolate the moment data (given the complexity of the issue, only the 2D case is treated in this paper, but using a method that generalizes to more variables).


nonlinear stochastic models, parametrization, Hankel and Jacobi matrices, dynamical system theory

Received 29 January 2020

Published 19 November 2020