Some limit properties for a hidden inhomogeneous Markov chain

Area of Science:

• Probability Theory
• Stochastic Processes
• Markov Chains

Background:

• Hidden Markov Models (HMMs) are widely used in various fields.
• Analyzing the asymptotic behavior of sums of random variables is crucial for understanding HMMs.
• Existing limit theorems may not fully capture the complexities of time-inhomogeneous and hidden states.

Purpose of the Study:

• To develop a general strong limit theorem for delayed sums of functions of random variables.
• To extend existing limit theorems to the specific context of hidden time inhomogeneous Markov chains (HTIMCs).
• To establish strong laws of large numbers as corollaries of the main theorem.

Main Methods:

• The study employs advanced techniques in probability theory and stochastic processes.
• A novel approach is used to handle the delayed sums and the time-inhomogeneous nature of the Markov chain.
• The derivation involves intricate mathematical analysis of the HTIMC structure.

Main Results:

• A general strong limit theorem for delayed sums of functions of random variables in HTIMCs is presented.
• The theorem provides a unified framework for analyzing such sums.
• Several strong laws of large numbers for HTIMCs are derived as direct consequences.

Conclusions:

• The established theorem offers a significant theoretical advancement for HTIMCs.
• The findings have implications for statistical inference and modeling in systems described by HTIMCs.
• This work provides a robust mathematical foundation for future research in this area.