Stochastic processes and their applications are a subfield of probability theory dealing with modeling real life data, representing the evolution of a system of random values over time. Familiar examples of stochastic processes include stock market fluctuations, signals such as audio and video, medical data such as a patient's blood pressure or temperature, and random movement, such as random walk. A generalization, the random field, describes evolution in time and space. Examples of random fields include static images, random terrain (landscapes), waves, or composition variations of a heterogeneous material. This research project is concerned with a better understanding of the degree of dependence in a stochastic process or random field by using new techniques for studying the interactions between its variables. These techniques are based on surprising relationships with concepts that appear in other fields of mathematics, such as algebra and dynamical systems. The results will lead to a more accurate prediction of future values of random evolution.
More specifically, in many situations, the interactions between the variables are viewed as a measure of the strength of dependence of a time series. The information about these interactions is given by a function called the "spectral density function," which plays a central role in the theory of time series and random fields. Other spectral notions in mathematics are: "operator spectral measure," which generates the transitions of a Markov process, and "empirical spectral distribution" of the eigenvalues of large random matrices with entries selected from random fields. This research project aims to establish several structural relationships that will build bridges between these three, apparently disparate notions, each having the word "spectral" in its definition. The results of this research will show that this is not a coincidence. As a matter of fact, they are interconnected and they shed light on one another. These relationships will facilitate determination of the strength of dependence in a random evolution. They will extend and improve tools used for analyzing stationary stochastic processes and fields by indicating new ways to estimate their spectral density. Some of the results will contribute to a better understanding of the limiting empirical eigenvalue distribution for random matrices with correlated entries, exhibiting long range dependence. The research aims to show that this limiting distribution is uniquely determined by the field's spectral density and to find a formula relating them.
