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Area of Science:

  • Probability Theory
  • Stochastic Processes
  • Ergodic Theory

Background:

  • Random walks on compact groups exhibit complex behavior, especially concerning convergence to stationary distributions.
  • Classical results for rational spans on finite cyclic subgroups rely on Markov chain theory.
  • Extending these to irrational spans requires new analytical approaches.

Purpose of the Study:

  • To extend central limit theorem and iterated logarithm law results for random walks on circle groups to irrational spans.
  • To describe the transition of Markov chains from finite to general state spaces.
  • To analyze the rate of weak convergence to the stationary distribution using Kolmogorov and total variation metrics.

Main Methods:

  • Analysis of random walks on the circle group with lattice steps.
  • Utilizing properties of finite state space Markov chains for rational spans.
  • Investigating the behavior of Markov chains as the span transitions from rational to irrational approximations.
  • Applying Kolmogorov and total variation metrics to assess convergence rates.

Main Results:

  • Established results for random walks with irrational spans on the circle group.
  • Explicit description of the transition from finite to general state space Markov chains.
  • Observed a phase transition in convergence rate (polynomial to exponential decay) in the rational case using the Kolmogorov metric.
  • Demonstrated purely exponential decay in the total variation metric.

Conclusions:

  • The behavior of random walks on circle groups is highly sensitive to the span, particularly the transition from rational to irrational values.
  • A novel phase transition phenomenon in convergence rates was identified in the rational case.
  • The study provides a comprehensive understanding of random walks on compact groups, bridging finite and infinite state space behaviors.