CS24: Recent Advances in Statistical Inference for Nonstationary Stochastic Processes

Organizer: Bartosz Majewski (AGH University of Krakow)

Statistical Properties of Oscillatory Processes with Stochastic Modulation in Amplitude and Time

Łukasz Lenart

This paper introduces a new semiparametric model for continuous nonstationary processes with irregular cyclicities. The proposed model consists of a superposition of cosines with a nonstationary phase shift process and a stationary amplitude process. The asymptotic properties of the first- and second-order moments of the considered process are investigated. Estimators of the asymptotic mean and autocovariance functions are introduced. The performance of the autocovariance function estimator is examined in a simulation study. Finally, the model is applied to a real biomedical dataset.

Spectral analysis of harmonizable processes with spectral mass concentrated on lines

Bartosz Majewski

Harmonizable processes are widely used to model nonstationary signals in fields such as economics, engineering, and medicine. They can be seen as a superposition of sine and cosine waves with random amplitudes. This representation allows us to analyze the dependency between their frequencies. In this context, spectral density and spectral coherence serve as frequency domain analogs of covariance and correlation, respectively.

In our research, we focus on harmonizable processes whose spectral measure is concentrated on a union of lines, potentially with non-unit slopes. This class of processes is a generalization of almost periodically correlated processes. It has practical applications in communication, particularly in the location of moving sources such as aircrafts, rockets, or hostile jamming emitters that transmit signals.

First, we address the spectral density estimation problem. We propose a periodogram frequency-smoothed along the support line as its estimator. We derive the asymptotic distribution of the rescaled estimator. Consequently, we obtain the asymptotic distribution of the rescaled spectral coherence estimator. In addition, we introduce a subsampling technique designed specifically for the class of processes considered. We establish its consistency and construct subsampling-based confidence intervals for the spectral characteristics of harmonizable processes. To illustrate the theoretical results, we present a simulation study for models commonly used in acoustics and communication.

Bibliography

\([1]\) Dudek, A. E. and Majewski, B. Asymptotic distribution and subsampling in spectral analysis for spectrally correlated processes. preprint 2024. https://hal.science/hal-04675084