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This study addresses the reachability problem for Markov chains, determining when a transition matrix can be achieved over time. It reviews solutions for time-homogeneous and time-inhomogeneous chains, suggesting future research on mixtures.

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Area of Science:

  • Mathematics
  • Probability Theory
  • Stochastic Processes

Background:

  • Empirical studies on labor mobility motivated research into Markov chain reachability.
  • The problem involves determining if a stochastic matrix P can be reached from the identity matrix I over a specific time period.

Purpose of the Study:

  • To summarize a complete answer to the Markov chain reachability problem for time-homogeneous cases.
  • To review diverse time-inhomogeneous cases studied since 1976.
  • To identify future research directions for mixtures of Markov chains.

Main Methods:

  • Utilized Kolmogorov forward and backward differential equations.
  • Analyzed time-homogeneous Markov chains.
  • Reviewed various time-inhomogeneous Markov chain models.

Main Results:

  • A complete solution for time-homogeneous Markov chains was established in 1976.
  • The paper reviews a range of solutions for time-inhomogeneous cases.
  • Identified the need for solutions to the reachability/embedding problem for mixtures of Markov chains.

Conclusions:

  • The reachability/embedding problem for Markov chains has been addressed for homogeneous and inhomogeneous cases.
  • Future research should focus on mixtures of Markov chains and their partial identifiability.
  • Understanding these conditions is crucial for analyzing complex systems with evolving transition probabilities.