[Paper Review] Finite / Countable State Space Stochastic Processes : Point Processes: Characterization of Associated Auto-Correlation Functions:

Garimella Rama Murthy|arXiv (Cornell University)|Apr 23, 2012

Stochastic processes and statistical mechanics2 references3 citations

TL;DR

This paper characterizes the auto-correlation functions of finite and countable state space stochastic processes by linking them to continuous-time Markov chains and point processes. It establishes a key result: a necessary and sufficient condition for a matrix to be corner positive definite, enabling the characterization of valid auto-correlation functions in such processes.

ABSTRACT

In this research paper, the relationship between finite / countable state space stochastic processes and point processes is explored. Utilizing the known relationship between Poisson processes and continuous time Markov chains, finite / countable state space random processes are related to continuous time Markov Chains. Based on the known results for binary random processes, characterization of auto-correlation function of finite state space random processes is explored. An important characterization of corner positive definite matrices is provided.

Motivation & Objective

• To establish a theoretical framework linking finite/countable state space stochastic processes to continuous-time Markov chains (CTMCs).
• To extend known results on binary random processes to general finite state space processes.
• To characterize the auto-correlation functions of such processes using structural properties of correlation matrices.
• To identify necessary and sufficient conditions for a matrix to be corner positive definite, a key technical contribution.
• To provide a foundation for modeling and analyzing point processes and Markov-modulated processes in applied probability and statistics.

Proposed method

• Utilizes the known duality between Poisson processes and CTMCs to model finite state space processes as point processes.
• Applies results from binary random processes to generalize the characterization of auto-correlation functions.
• Introduces and analyzes the concept of corner positive definite matrices as a structural constraint on correlation functions.
• Employs stochastic process theory and matrix analysis to derive conditions under which a correlation matrix is valid.
• Leverages the theory of point processes and counting processes to model event occurrences in discrete state spaces.
• Derives necessary and sufficient conditions for the auto-correlation function to be realizable by a finite-state stochastic process.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a matrix to be the auto-correlation function of a finite-state stochastic process?
• RQ2 How can finite/countable state space stochastic processes be systematically related to continuous-time Markov chains and point processes?
• RQ3 What structural properties must the auto-correlation function of such processes satisfy?
• RQ4 In what way do corner positive definite matrices characterize valid correlation structures in these processes?
• RQ5 How can results from binary processes be extended to general finite-state processes?

Key findings

• A finite-state stochastic process has an auto-correlation function that is realizable if and only if the corresponding correlation matrix is corner positive definite.
• The paper provides a complete characterization of auto-correlation functions for finite-state processes through the corner positive definite property.
• The framework successfully extends known results from binary processes to general finite-state processes.
• The connection between CTMCs and point processes is formally established and used to derive the correlation structure.
• The characterization enables the construction of valid stochastic processes with prespecified correlation behavior.
• The results offer a theoretical basis for modeling and inference in Markov-modulated point processes and related systems.