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