Estimation of certain parameters of a stationary hybrid process involving a time series and a point process

A method is presented for estimating the cross-spectral density of a hybrid process involving a time series and a point process. The method is based on the generalized cross-periodogram statistic, which is smoothed by splitting the whole record of the data into a number of disjoint subrecords. Estimates of the coherence function and the cross-covariance function can also be obtained by using the estimate of the cross-spectral density. The distribution of the cross-covariance function between a time series and a point process is shown to be asymptotically normal. The theoretical results are used in the study of a complex physiological system. It is shown that the presence of a gamma motor neuron (gamma stimulation) modifies the effect of the length changes on the complex system at low frequencies (the length changes, and the response of the system become uncorrelated in the range 3-30 Hz) while the effect remains unchanged at higher frequencies. As a comparison it is shown that the presence of the length changes weakens the effect of the gamma stimulation on the complex system.