Zero-one law for the rates of convergence in the Birkhoff ergodic theorem with continuous time

2021 ◽  
Vol 24 (2) ◽  
pp. 65-80
Author(s):  
A. G. Kachurovskii ◽  
I. V. Podvigin ◽  
A. A. Svishchev
Keyword(s):  
Ergodic Theorem ◽  
Continuous Time ◽  
Rates Of Convergence ◽  
Birkhoff Ergodic Theorem

scholarly journals Noncommutative weighted individual ergodic theorems with continuous time

2020 ◽  
Vol 23 (02) ◽  
pp. 2050013
Author(s):  
Vladimir Chilin ◽  
Semyon Litvinov
Keyword(s):  
Ergodic Theorem ◽  
Continuous Time ◽  
Symmetric Spaces ◽  
Von Neumann Algebra ◽  
Ergodic Theorems ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Von Neumann ◽  
Marcinkiewicz Spaces ◽  
Local Ergodic Theorem

We show that ergodic flows in the noncommutative [Formula: see text]-space (associated with a semifinite von Neumann algebra) generated by continuous semigroups of positive Dunford–Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly. The corresponding local ergodic theorem is also proved. We then extend these results to arbitrary noncommutative fully symmetric spaces and present applications to noncommutative Orlicz (in particular, noncommutative [Formula: see text]-spaces), Lorentz, and Marcinkiewicz spaces. The commutative counterparts of the results are derived.

scholarly journals Subgeometric rates of convergence for a class of continuous-time Markov process

2005 ◽  
Vol 42 (03) ◽  
pp. 698-712
Author(s):  
Zhenting Hou ◽  
Yuanyuan Liu ◽  
Hanjun Zhang
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Invariant Probability Measure ◽  
Sufficient Conditions ◽  
Rates Of Convergence ◽  
Transition Function ◽  
Total Variation Norm ◽  
Rate Functions ◽  
General State Space ◽  
Explicit Bounds

Let (Φ t ) t∈ℝ+ be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure π. We investigate the rates of convergence of the transition function P t (x, ·) to π; specifically, we find conditions under which r(t)||P t (x, ·) − π|| → 0 as t → ∞, for suitable subgeometric rate functions r(t), where ||·|| denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

Exponential Rates of Convergence in the Ergodic Theorem: A Constructive Approach

2010 ◽  
Vol 139 (3) ◽  
pp. 367-374
Author(s):  
G. G. Bosco ◽  
F. P. Machado ◽  
Thomas Logan Ritchie
Keyword(s):  
Ergodic Theorem ◽  
Rates Of Convergence ◽  
Constructive Approach ◽  
Exponential Rates
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