Cumulant methods in the estimation of response functions in time-invariant linear systems

TL;DR

This paper explores cumulant analysis for estimating impulse response functions in continuous time-invariant linear systems, including noisy systems, demonstrating asymptotic normality of estimators through integral representations and convergence proofs.

Contribution

It introduces a cumulant-based approach for response function estimation in linear systems, extending analysis to systems with noise and proving estimator asymptotic normality.

Findings

Proved convergence of integral representations of cumulants to zero.

Established asymptotic normality of cross-correlogram estimators.

Applied cumulant analysis to systems with inner noise.

Abstract

Thesis is devoted to the application of cumulant analysis in the estimation of impulse response functions for continuous time-invariant linear systems, including systems with inner noises. The main assumption of the work is the second-order integration of the impulse response function. Our study deals with cumulant analysis of sample cross-correlograms between stationary Gaussian stochastic processes. An important role was played by integral representations for the higher-order cumulants of these second-order statistics. Using the diagram formula, all representations are reduced to the finite sums of integrals involving cyclic products of kernels. In the work we proved the convergence to zero of the corresponding integrals. Then, since the Gaussian distribution is uniquelly determined by its cumulants and also all higher-order cumulants of the estimators tend to zero, we establish the asymptotic normality of the integral-type cross-correlogram estimators.