A Note on the Brikhoff Ergodic Theorem Nikola Sandri c University - PowerPoint PPT Presentation

A Note on the Brikhoff Ergodic Theorem Nikola Sandri´ c University of Zagreb Probability and Analysis B˛ edlewo May 15-19, 2017 Nikola Sandri´ c Probability and Analysis May 15, 2017 1 / 19

Outline Motivation and Preliminaries 1 Main results 2 Examples 3 Nikola Sandri´ c Probability and Analysis May 15, 2017 2 / 19

Motivation and preliminaries Let M = (Ω , F , { P x } x ∈ S , {F t } t ∈ T , { θ t } t ∈ T , { M t } t ∈ T ) be a Markov process with state space ( S , S ) . Here, T is the time set Z + or R + . Nikola Sandri´ c Probability and Analysis May 15, 2017 3 / 19

Motivation and preliminaries Let M = (Ω , F , { P x } x ∈ S , {F t } t ∈ T , { θ t } t ∈ T , { M t } t ∈ T ) be a Markov process with state space ( S , S ) . Here, T is the time set Z + or R + . A measure π ( dy ) on S is said to be invariant for M if � p t ( x , dy ) π ( dx ) = π ( dy ) , t ∈ T . S Nikola Sandri´ c Probability and Analysis May 15, 2017 3 / 19

Motivation and preliminaries Let M = (Ω , F , { P x } x ∈ S , {F t } t ∈ T , { θ t } t ∈ T , { M t } t ∈ T ) be a Markov process with state space ( S , S ) . Here, T is the time set Z + or R + . A measure π ( dy ) on S is said to be invariant for M if � p t ( x , dy ) π ( dx ) = π ( dy ) , t ∈ T . S A set B ∈ F is said to be shift-invariant (for M ) if θ − 1 B = B for all t ∈ T . t The shift-invariant σ -algebra I is a collection of all such shift-invariant sets. Nikola Sandri´ c Probability and Analysis May 15, 2017 3 / 19

Motivation and preliminaries Theorem (Birkhoff ergodic theorem) Let M be a Markov process with invariant probability measure π ( dy ) . Then, for any f ∈ L p ( S , π ) , p ≥ 1, the following limit holds 1 � P π -a.s. and in L p (Ω , P π ) , lim f ( M s ) τ ( ds ) = E π [ f ( M 0 ) |I ] t t →∞ [ 0 , t ) where τ ( dt ) is the counting measure when T = Z + and Lebesgue measure when T = R + . Nikola Sandri´ c Probability and Analysis May 15, 2017 4 / 19

Motivation and preliminaries A Markov process M is said to be ergodic if it possesses an invariant probability measure π ( dy ) and if I is trivial with respect to P π ( d ω ) , that is, P π ( B ) = 0 or 1 for every B ∈ I . Nikola Sandri´ c Probability and Analysis May 15, 2017 5 / 19

Motivation and preliminaries A Markov process M is said to be ergodic if it possesses an invariant probability measure π ( dy ) and if I is trivial with respect to P π ( d ω ) , that is, P π ( B ) = 0 or 1 for every B ∈ I . In addition to the assumptions of the Birkhoff ergodic theorem, if M is ergodic then we conclude 1 � � P π -a.s. and in L p (Ω , P π ) . lim f ( M s ) τ ( ds ) = f ( y ) π ( dy ) t t →∞ [ 0 , t ) S Nikola Sandri´ c Probability and Analysis May 15, 2017 5 / 19

Motivation and preliminaries A Markov process M is said to be ergodic if it possesses an invariant probability measure π ( dy ) and if I is trivial with respect to P π ( d ω ) , that is, P π ( B ) = 0 or 1 for every B ∈ I . In addition to the assumptions of the Birkhoff ergodic theorem, if M is ergodic then we conclude 1 � � P π -a.s. and in L p (Ω , P π ) . lim f ( M s ) τ ( ds ) = f ( y ) π ( dy ) t t →∞ [ 0 , t ) S Question: Can we conclude the above relation for any initial distribution of M ? Nikola Sandri´ c Probability and Analysis May 15, 2017 5 / 19

Motivation and preliminaries Meyn and Tweedie (2009) have shown that the following are equivalent: (a) the above relation holds P µ -a.s. for any f ∈ L p ( S , π ) and any µ ∈ P ( S ) (b) M is a positive Harris recurrent Markov process. Nikola Sandri´ c Probability and Analysis May 15, 2017 6 / 19

Motivation and preliminaries Meyn and Tweedie (2009) have shown that the following are equivalent: (a) the above relation holds P µ -a.s. for any f ∈ L p ( S , π ) and any µ ∈ P ( S ) (b) M is a positive Harris recurrent Markov process. A Markov process M is called ϕ -irreducible if for the σ -finite measure ϕ ( dy ) on S , ϕ ( B ) > 0 implies � p t ( x , B ) τ ( dt ) > 0 , x ∈ S . T Nikola Sandri´ c Probability and Analysis May 15, 2017 6 / 19

Motivation and preliminaries The process M is called Harris recurrent if it is ϕ -irreducible, and ϕ ( B ) > 0 implies � P x -a.s. for all x ∈ S . 1 { M t ∈ B } τ ( dt ) = ∞ T Nikola Sandri´ c Probability and Analysis May 15, 2017 7 / 19

Motivation and preliminaries The process M is called Harris recurrent if it is ϕ -irreducible, and ϕ ( B ) > 0 implies � P x -a.s. for all x ∈ S . 1 { M t ∈ B } τ ( dt ) = ∞ T It is well known that every Harris recurrent Markov process admits a unique (up to constant multiplies) invariant (not necessary probability) measure. If the invariant measure is finite, then the process is called positive Harris recurrent; otherwise it is called null Harris recurrent. Nikola Sandri´ c Probability and Analysis May 15, 2017 7 / 19

Motivation and preliminaries If M is also aperiodic, Meyn and Tweedie (2009) have proved that (a), (b) and (c) are equivalent to (d) M is strongly ergodic. Nikola Sandri´ c Probability and Analysis May 15, 2017 8 / 19

Motivation and preliminaries If M is also aperiodic, Meyn and Tweedie (2009) have proved that (a), (b) and (c) are equivalent to (d) M is strongly ergodic. A Markov process M is said to be strongly ergodic if there exists π ∈ P ( S ) such that t →∞ d TV ( p t ( x , dy ) , π ( dy )) = 0 , lim x ∈ S , where d TV denotes the total variation metric on P ( S ) given by � � d TV ( µ ( dy ) , ν ( dy )) := 1 � � � � sup f ( y ) µ ( dy ) − f ( y ) ν ( dy ) � . � � 2 f ∈ B b ( S ) , | f | ∞ ≤ 1 � S S Nikola Sandri´ c Probability and Analysis May 15, 2017 8 / 19

Motivation and preliminaries In the discrete-time case, Hernandez-Lerma and Lasserre (2000) have shown that if M has a unique invariant probability measure π ( dy ) , then either t →∞ d TV ( p t ( x , dy ) , π ( dy )) = 0 π -a.e., or (i) lim (ii) π ( dy ) ⊥ � ∞ t = 1 p t ( x , dy ) π -a.e. and p t ( x , dy ) converges weakly to π ( dy ) . Nikola Sandri´ c Probability and Analysis May 15, 2017 9 / 19

Motivation and preliminaries In the discrete-time case, Hernandez-Lerma and Lasserre (2000) have shown that if M has a unique invariant probability measure π ( dy ) , then either t →∞ d TV ( p t ( x , dy ) , π ( dy )) = 0 π -a.e., or (i) lim (ii) π ( dy ) ⊥ � ∞ t = 1 p t ( x , dy ) π -a.e. and p t ( x , dy ) converges weakly to π ( dy ) . Goal: To relax the notion of strong ergodicity and, under these new assumptions, conclude a version of the Birkhoff ergodic theorem which holds for any initial distribution of the process. Nikola Sandri´ c Probability and Analysis May 15, 2017 9 / 19

Main results In the sequel we assume that ( S , S ) is a Polish space with bounded (say by 1) metric d . In particular, Lip ( S ) ⊆ C b ( S ) . Nikola Sandri´ c Probability and Analysis May 15, 2017 10 / 19

Main results In the sequel we assume that ( S , S ) is a Polish space with bounded (say by 1) metric d . In particular, Lip ( S ) ⊆ C b ( S ) . Wasserstein metric of order one on P ( S ) is defined by � � � � � � d W ( µ ( dy ) , ν ( dy )) := sup f ( y ) µ ( dy ) − f ( y ) ν ( dy ) � . � � � S S f ∈ Lip ( S ) , | f | Lip ≤ 1 Nikola Sandri´ c Probability and Analysis May 15, 2017 10 / 19

Main results In the sequel we assume that ( S , S ) is a Polish space with bounded (say by 1) metric d . In particular, Lip ( S ) ⊆ C b ( S ) . Wasserstein metric of order one on P ( S ) is defined by � � � � � � d W ( µ ( dy ) , ν ( dy )) := sup f ( y ) µ ( dy ) − f ( y ) ν ( dy ) � . � � � S S f ∈ Lip ( S ) , | f | Lip ≤ 1 Recall, d W ≤ d TV , and d W metrizes the weak convergence of probability measures. More precisely, { µ n } n ∈ N ⊆ P ( S ) converges to µ ∈ P ( S ) with respect to d W if, and only if, � � lim f ( y ) µ n ( dy ) = f ( y ) µ ( dy ) , f ∈ C b ( S ) . n →∞ S S Nikola Sandri´ c Probability and Analysis May 15, 2017 10 / 19

Main results Theorem Assume that there is π ∈ P ( S ) satisfying � d W ( p t ( y , dz ) , π ( dz )) p s ( x , dy ) = 0 , t →∞ sup lim x ∈ S . (1) s ∈ T S Then, for any p ≥ 1, f ∈ Lip ( S ) and µ ∈ P ( S ) , 1 � � L p (Ω , P µ ) f ( M s ) τ ( ds ) − − − − − → f ( y ) π ( dy ) , (2) t t ր∞ [ 0 , t ) S L p (Ω , P µ ) denotes the convergence in L p (Ω , P µ ) . where − − − − − → t ր∞ Nikola Sandri´ c Probability and Analysis May 15, 2017 11 / 19

Main results Corollary Assume the assumptions from the previous theorem. Then, (i) (2) holds for all f ∈ C c ( S ) and µ ∈ P ( S ) . In particular, if S is compact, (2) holds for all f ∈ C ( S ) and µ ∈ P ( S ) . (ii) provided ( S , d ) is locally compact, (2) holds for all f ∈ C ∞ ( S ) and µ ∈ P ( S ) . (iii) provided π ( dy ) is an invariant measure for M and ( S , d ) is locally compact, 1 � � L p (Ω , P π ) p ≥ 1 , f ∈ L p ( S , π ) . f ( M s ) τ ( ds ) − − − − − → f ( y ) π ( dy ) , t t ր∞ [ 0 , t ) S Nikola Sandri´ c Probability and Analysis May 15, 2017 12 / 19