We characterize global stability of monotone Markov processes, showing that any such process has a unique, globally stable stationary distribution if and only if it is asymptotically contractive and has a tight trajectory. The characterization follows from a new fixed point theorem for order-preserving maps on preordered metric spaces, and covers compact and noncompact state spaces and both discrete and continuous time. We illustrate our results with applications to Bayesian learning and income dynamics.
In this article, stable processes with different types of constraints on the given domain D⊂Rn are studied. We first establish the convergence results for the infinitesimal generators of reflecting stable processes, and prove that reflecting compound stable processes converge to the reflecting Brownian motion on D in the sense of 1-Wasserstein distance as α↑2. Secondly, we look into the difference between the generators of censored α-stable processes and killed Brownian motions and observe the divergence of compound censored stable processes in 1-Wasserstein distance. Finally, we study the convergence of first boundary exit distributions for reflecting and censored stable processes. Explicit bounds for heat kernels and Green functions play an important role when discussing the convergence results as α↑2.
We establish upper large deviation bounds for noncommutative random variables and martingales. For independent mean-zero noncommutative random variables, we characterize the uniform exponential integrability in terms of Bernstein-type deviation inequality for finite sums. For noncommutative martingale differences, we prove upper deviation estimates under uniform exponential integrability and uniform Lp-boundedness assumptions, respectively. Finally, we establish a noncommutative version of Gordin’s decomposition and combine it with the martingale large deviation inequality to derive convergence rates in nocommutative ergodic theory.
We construct new classes of examples of self-similar processes with stationary increments indexed by Qp via stable integrals. Classical constructions arise from the real counterpart and from discounted branching random walks. We discuss a new decoration technique that significantly enlarges these classes. The decoration technique makes use of the special symmetry of Qp to obtain self-similarity and stationarity of increments, and it does not have an analogue on the real line. We also show that these enlarged classes of decorated processes are distribution-wise pairwise incomparable under inclusion.
For Zd actions equipped with a completely commuting filtration, normalized partial sums of martingale differences converge in distribution (see [1]) but even if the action is ergodic, the limit law need not be normal. Explication of this phenomenon has been studied in [2] but often in dimension 2 only. It turned out that the non normality is related to the factor of product type I.Here we extend the necessary and sufficient condition of normality from dimension 2 to all finite dimensions, as well as a sufficient condition of normality for martingale differences in the orthocomplement of L2(I). At the same time an improved and corrected version of the central limit theorem for random fields from [1] with dimension d > 2 is given.
