Convergence theorems for barycentric maps Fumio Hiai Tohoku - PowerPoint PPT Presentation

Convergence theorems for barycentric maps Fumio Hiai Tohoku University 2018, July (at Be ¸dlewo) Joint work 1 with Yongdo Lim 1 F.H. and Y. Lim, Convergence theorems for contractive barycentric maps, arXiv:1805.08558 [math.PR]. Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 1 / 24

Idea When ( M , d ) is a global NPC = CAT(0) space, martingale convergence, strong law of large numbers and ergodic theorem were developed for M -valued random variables by Es-Sahib and Heinich, Sturm, Austin, Navas, ...... By using the disintegration theorem, we develop those stochastic convergence theorems when ( M , d ) is a general complete metric space with a contractive barycentric map β . E.g., M = P ( H ) is the positive invertible operators on a Hilbert space H , d = d T is the Thompson metric, and β is the Cartan barycenter (Karcher mean). Plan Conditional expectations Martingale convergence theorem Ergodic theorem Large deviation principle Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 2 / 24

Preliminaries ( M , d ) is a complete metric space with the Borel σ -algebra B ( M ) . P ( M ) is the set of probability measures on B ( M ) with full support. For 1 ≤ p < ∞ , P p ( M ) is the set of µ ∈ P ( M ) such that ∫ M d p ( x , y ) d µ ( y ) < ∞ for some (hence, all) x ∈ M . P 1 ( M ) ⊃ P p ( M ) ⊃ P q ( M ) , 1 < p < q < ∞ . For 1 ≤ p < ∞ , the p -Wasserstein distance is ∫ [ ] 1 / p d W d p ( x , y ) d π ( x , y ) p ( µ, ν ) : = inf , µ, ν ∈ P ( M ) , π ∈ Π ( µ,ν ) M × M where Π ( µ, ν ) is the set of π ∈ P ( M × M ) whose marginals are µ, ν . d W 1 ≤ d W p ≤ d W q , 1 < p < q < ∞ , and ( P p ( M ) , d W p ) is a complete metric space. Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 3 / 24

( Ω , A , P) is a probability space. For 1 ≤ p < ∞ , L p ( Ω ; M ) = L p ( Ω , A , P; M ) is the set of strongly measurable functions f : Ω → M such that ∫ Ω d p ( x , f ( ω )) d P( ω ) < ∞ for some (hence, all) x ∈ M . L 1 ( Ω ; M ) ⊃ L p ( Ω ; M ) ⊃ L q ( Ω ; M ) 1 < p < q < ∞ . Lemma Let 1 ≤ p < ∞ . L p ( Ω ; M ) is a complete metric space with the L p -distance [∫ ] 1 / p d p ( ϕ ( ω ) , ψ ( ω )) d P( ω ) d p ( ϕ, ψ ) : = . Ω If ϕ ∈ L p ( Ω ; M ) , then the push-forward measure ϕ ∗ P ∈ P p ( M ) . If ϕ, ψ ∈ L p ( Ω ; M ) , then d W p ( ϕ ∗ P , ψ ∗ P) ≤ d p ( ϕ, ψ ) . Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 4 / 24

Conditional expectations Conditional expectations Let 1 ≤ p < ∞ be fixed, and assume that β : P p ( M ) → M is a p -contractive barycentric map, i.e., β ( δ x ) = x for all x ∈ M and d ( β ( µ ) , β ( ν )) ≤ d W µ, ν ∈ P p ( M ) . p ( µ, ν ) , Definition The β -expectation E β ( ϕ ) of ϕ ∈ L p ( Ω ; M ) is defined by E β ( ϕ ) : = β ( ϕ ∗ P) ∈ M . Proposition d ( E β ( ϕ ) , E β ( ψ )) ≤ d p ( ϕ, ψ ) for ϕ, ψ ∈ L p ( Ω ; M ) . E β (1 Ω x ) = x for x ∈ M . Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 5 / 24

Conditional expectations Next, assume that ( Ω , A ) is a standard Borel space, i.e., isomorphic to ( X , B ( X )) of a Polish space X . Let B be a sub- σ -algebra of A . Then there exists a disintegration (P ω ) ω ∈ Ω with respect to B , a family of probability measures on ( Ω , A ) , such that for every A ∈ A , (i) ω ∈ Ω �→ P ω ( A ) is B -measurable, (ii) ω �→ P ω ( A ) is a conditional expectation E B (1 A ) of 1 A with respect to B , Such a family (P ω ) ω ∈ Ω is unique up to a P -null set, and moreover (iii) for every f ∈ L 1 ( Ω ; R ) , f ∈ L 1 ( Ω , A , P ω ; R ) for P -a.e. ω and ∫ ω �→ Ω f ( τ ) d P ω ( τ ) is a conditional expectation E B ( f ) of f with respect to B . In particular, ∫ ∫ [∫ ] f d P = f ( τ ) d P ω ( τ ) d P( ω ) . Ω Ω Ω Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 6 / 24

Conditional expectations Definition The β -conditional expectation E β B ( ϕ ) of ϕ ∈ L p ( Ω ; M ) is defined by E β ω ∈ Ω . B ( ϕ ) : = β ( ϕ ∗ P ω ) , Theorem Let ϕ, ψ ∈ L p ( Ω ; M ) . (1) E β B ( ϕ ) ∈ L p ( Ω , B , P; M ) . (2) d p ( E β B ( ϕ ) , E β B ( ψ )) ≤ d p ( ϕ, ψ ) . (3) ϕ ∈ L p ( Ω , B , P; M ) if and only if E β B ( ϕ ) = ϕ . Hence E β B ( E β B ( ϕ )) = E β B ( ϕ ) . (4) When B = {∅ , Ω } , E β B ( ϕ ) = E β ( ϕ ) . Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 7 / 24

Conditional expectations When ( M , d ) is a global NPC space or CAT(0) space, (i.e., for any x 0 , x 1 ∈ M there exists a y ∈ M such that d 2 ( y , z ) ≤ d 2 ( x 0 , z ) + d 2 ( x 1 , z ) − d 2 ( x 0 , x 1 ) for all z ∈ M ) , 2 4 the canonical barycentric map λ on P 1 ( M ) is ∫ [ d 2 ( z , x ) − d 2 ( y , x )] d µ ( x ) , µ ∈ P 1 ( M ) , λ ( µ ) : = arg min z ∈ M M independently of the choice of y ∈ M . Sturm’s 2 definition in the case of a global NPC space is E B ( ϕ ) : = arg min d 2 ( ϕ, ψ ) ψ ∈ L 2 ( Ω , B , P; M ) for ϕ ∈ L 2 ( Ω ; M ) , and E B extends continuously to L 1 ( Ω ; M ) . 2 K.-T. Sturm, Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature, Ann. Probab. 30 (2002), 1195–1222. Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 8 / 24

Conditional expectations Theorem Assume that ( Ω , A ) is a standard Borel space and ( M , d ) is a global NPC space. Then for every p ∈ [1 , ∞ ) and ϕ ∈ L p ( Ω ; M ) , E B ( ϕ ) = E λ B ( ϕ ) . Remark Unlike the usual conditional expectation, the β -conditional expectation is not associative in general, that is, for sub- σ -algebras C ⊂ B ⊂ A , E β C ( E β B ( ϕ )) � E β C ( ϕ ) . Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 9 / 24

Martingale convergence theorem Martingale convergence theorem Let ( Ω , A , P) be a standard Borel probability space, and {B n } ∞ n = 1 be a sequence of sub- σ -algebras of A such that B 1 ⊂ B 2 ⊂ · · · or B 1 ⊃ B 2 ⊃ · · · . Let B ∞ be the sub- σ -algebra generated by ∪ ∞ n = 1 B n or B ∞ : = ∩ ∞ n = 1 B n . Theorem Assume that ( Ω , A , P) and {B n } ∞ n = 1 are as stated above. Let β : P p ( M ) → M be as before. Then for every ϕ ∈ L p ( Ω ; M ) , as n → ∞ , d p ( E β B ∞ ( ϕ ) ) − B n ( ϕ ) , E β → 0 , d ( E β B ∞ ( ϕ )( ω ) ) − B n ( ϕ )( ω ) , E β → 0 a.e. Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 10 / 24

Martingale convergence theorem Assume that B 1 ⊂ B 2 ⊂ · · · . Since E β B m ( E β B n ( ϕ )) = E β B m ( ϕ ) ( m < n ) does not hold, we follow Sturm’s 2 idea to define martingales of M -valued random variables. Definition For ϕ ∈ L p ( Ω ; M ) and k ≥ 1 , we can define E β [ ϕ ∥ ( B n ) n ≥ k ] : = lim m →∞ E β B k ◦ · · · ◦ E β B m ( ϕ ) m →∞ E β B k ◦ · · · ◦ E β B m ( E β = lim B ∞ ϕ )) in metric d p . Call E β [ ϕ ∥ ( B n ) n ≥ k ] the filtered β -conditional expectation with respect to ( B n ) n ≥ k . Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 11 / 24

Martingale convergence theorem Proposition Let ϕ, ψ ∈ L p ( Ω ; M ) . (1) E β [ ϕ ∥ ( B n ) n ≥ k ] ∈ L p ( Ω , B k , P; M ) for all k ≥ 1 . (2) For every k ≥ 1 , ϕ ∈ L p ( Ω , B k , P; M ) if and only if E β [ ϕ ∥ ( B n ) n ≥ k ] = ϕ . (3) d p ( E β [ ϕ ∥ ( B n ) n ≥ k ] , E β [ ψ ∥ ( B n ) n ≥ k ]) ≤ d p ( ϕ, ψ ) for all k ≥ 1 . (4) Associativity: For every l ≥ k ≥ 1 , E β [ E β [ ϕ ∥ ( B n ) n ≥ l ] ∥ ( B n ) n ≥ k ] = E β [ ϕ ∥ ( B n ) n ≥ k ] . Definition A sequence { ϕ k } ∞ k = 1 in L p ( Ω ; M ) is called a filtered β -martingale with respect to {B n } ∞ n = 1 if ϕ k ∈ L p ( Ω , B k , P; M ) for every k ≥ 1 and E β [ ϕ k + 1 ∥ ( B n ) n ≥ k ] = ϕ k , k ≥ 1 , equivalently, E β [ ϕ l ∥ ( B n ) n ≥ k ] = ϕ k for all l ≥ k ≥ 1 . Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 12 / 24

Martingale convergence theorem Theorem Let { ϕ k } ∞ k = 1 be a filtered β -martingale with respect to {B n } . Then the following are equivalent: (i) there exists a ϕ ∈ L p ( Ω ; M ) such that ϕ k = E β [ ϕ ∥ ( B n ) n ≥ k ] for all k ≥ 1; (ii) ϕ k converges to some ϕ ∞ ∈ L p ( Ω , B ∞ , P; M ) in metric d p as k → ∞ . Remark Assume that ( M , d ) is a global NPC space (or more generally, a complete length space) and it is locally compact. It is known 2 that if { ϕ k } in L p ( Ω ; M ) is a filtered martingale and sup k d p ( z , ϕ k ) < ∞ for some z ∈ M , then there exists a B ∞ -measurable function ϕ ∞ : Ω → M such that ϕ k ( ω ) → ϕ ∞ ( ω ) P -a.e. But it is unknown that this holds in our general setting. Fumio Hiai (Tohoku University) Convergence theorems 2018, July (at Be ¸dlewo) 13 / 24