A monotone Markov process has a globally stable stationary distribution if and only if it is asymptotically contractive and has a tight trajectory.
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Markov processes describe how a system's state changes randomly over time. Many economic models use monotone versions where higher current states lead to higher future states on average. Proving these settle into a unique stable long-run distribution has previously required different sufficient conditions for each model, especially when states can grow without bound. The paper shows that two properties together are both necessary and sufficient for global stability: the process must be asymptotically contractive, meaning distances between possible future paths shrink over time, and trajectories must be tight, meaning the state does not escape to infinity with positive probability. The result covers compact and noncompact spaces, discrete and continuous time, and nonlinear operators that depend on aggregate variables. Applications include wage dynamics, Bayesian learning with shocks, and income processes producing Pareto tails.
Core claim
a monotone Markov process has a globally stable stationary distribution if and only if it is asymptotically contractive and has a tight trajectory
Load-bearing premise
The Markov process is monotone and the specific definitions of asymptotic contractiveness and tight trajectories apply to the given state space and operator.
read the original abstract
Many economic models feature monotone Markov dynamics on state spaces that may be noncompact. Establishing existence, uniqueness, and stability of stationary distributions in such settings has required a patchwork of sufficient conditions, each tailored to specific applications. We provide a single necessary and sufficient condition: a monotone Markov process has a globally stable stationary distribution if and only if it is asymptotically contractive and has a tight trajectory. This characterization covers both compact and noncompact state spaces, discrete and continuous time, and extends to nonlinear Markov operators that depend on aggregate state. We demonstrate the result through applications to wage dynamics, Bayesian learning with belief shocks, and income processes that generate Pareto tails.
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The central claim rests on the monotonicity assumption and the newly introduced notions of asymptotic contractiveness and tight trajectories, which are not further detailed in the abstract.
axioms (1)
domain assumptionThe Markov process is monotone Stated explicitly in the title and abstract as the class of processes under consideration.
invented entities (2)
asymptotically contractiveno independent evidence purpose: Condition ensuring distances between trajectories contract over time Introduced as one of the two necessary and sufficient properties in the characterization.
tight trajectoryno independent evidence purpose: Condition ensuring trajectories do not escape to infinity in probability Introduced as the second necessary and sufficient property in the characterization.
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