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Subexponential lower bounds for f-ergodic Markov processes.

Miha Brešar1, Aleksandar Mijatović1

  • 1Department of Statistics, University of Warwick, Coventry, UK.

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Summary
This summary is machine-generated.

Researchers developed a new criterion to establish lower bounds on Markov process convergence rates. This method uses novel martingale conditions and path-wise arguments for analyzing invariant measure tails and convergence speed.

Keywords:
Invariant measureLower boundsLyapunov functions and Foster-type drift conditionsRate of convergence in f-variation and total variationReturn timesSubgeometric ergodicity

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Area of Science:

  • Probability Theory
  • Stochastic Processes
  • Ergodic Theory

Background:

  • Markov processes are fundamental in modeling dynamic systems.
  • Understanding convergence rates to invariant measures is crucial for analyzing system long-term behavior.
  • Existing methods primarily focus on upper bounds, leaving lower bounds less explored.

Purpose of the Study:

  • To introduce a novel criterion for establishing lower bounds on the rate of convergence in f-variation for continuous-time ergodic Markov processes.
  • To develop a general approach for proving lower bounds on invariant measure tails and convergence rates.
  • To provide a method analogous to Lyapunov drift conditions for upper bounds.

Main Methods:

  • Development of super- and submartingale conditions for specific Markov process functionals.
  • Utilizing path-wise arguments to derive lower bounds on excursion heights and durations from bounded sets.
  • Applying the criterion to elliptic diffusions and Lévy-driven stochastic differential equations.

Main Results:

  • A new criterion for lower bounds on convergence rates in f-variation is established.
  • The method successfully derives lower bounds on invariant measure tails.
  • Applied to specific models, the derived lower bounds asymptotically match known upper bounds, confirming convergence rates.

Conclusions:

  • The developed criterion offers a general approach for proving lower bounds on Markov process convergence.
  • The methodology, using path-wise arguments and martingale conditions, is broadly applicable.
  • This work complements existing methods for analyzing convergence rates and is expected to find wide application.