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

1

Motivation and Preliminaries

2

Main results

3

Examples

Nikola Sandri´ c Probability and Analysis May 15, 2017 2 / 19

Motivation and preliminaries Let M = (Ω, F, {Px}x∈S, {Ft}t∈T, {θt}t∈T, {Mt}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, {Px}x∈S, {Ft}t∈T, {θt}t∈T, {Mt}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

• S

pt(x, dy)π(dx) = π(dy), t ∈ T.

Nikola Sandri´ c Probability and Analysis May 15, 2017 3 / 19

Motivation and preliminaries Let M = (Ω, F, {Px}x∈S, {Ft}t∈T, {θt}t∈T, {Mt}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

• S

pt(x, dy)π(dx) = π(dy), t ∈ T. A set B ∈ F is said to be shift-invariant (for M) if θ−1

t

B = B for all 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 ∈ Lp(S, π), p ≥ 1, the following limit holds lim

t→∞

1 t

• [0,t)

f(Ms)τ(ds) = Eπ[f(M0)|I] Pπ-a.s. and in Lp(Ω, Pπ), 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 Pπ(dω), that is, Pπ(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 Pπ(dω), that is, Pπ(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 lim

t→∞

1 t

• [0,t)

f(Ms)τ(ds) =

• S

f(y)π(dy) Pπ-a.s. and in Lp(Ω, Pπ).

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 Pπ(dω), that is, Pπ(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 lim

t→∞

1 t

• [0,t)

f(Ms)τ(ds) =

• S

f(y)π(dy) Pπ-a.s. and in Lp(Ω, Pπ). 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 Pµ-a.s. for any f ∈ Lp(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 Pµ-a.s. for any f ∈ Lp(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

• T

pt(x, B)τ(dt) > 0, x ∈ S.

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

• T

1{Mt∈B}τ(dt) = ∞ Px-a.s. for all x ∈ S.

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

• T

1{Mt∈B}τ(dt) = ∞ Px-a.s. for all x ∈ S. 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 lim

t→∞ dTV(pt(x, dy), π(dy)) = 0,

x ∈ S, where dTV denotes the total variation metric on P(S) given by dTV(µ(dy), ν(dy)) := 1 2 sup

f∈Bb(S), |f|∞≤1

• S

f(y)µ(dy) −

• S

f(y)ν(dy)

• .

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 (i) lim

t→∞ dTV(pt(x, dy), π(dy)) = 0 π-a.e., or

(ii) π(dy) ⊥ ∞

t=1 pt(x, dy) π-a.e. and pt(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 (i) lim

t→∞ dTV(pt(x, dy), π(dy)) = 0 π-a.e., or

(ii) π(dy) ⊥ ∞

t=1 pt(x, dy) π-a.e. and pt(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) ⊆ Cb(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) ⊆ Cb(S). Wasserstein metric of order one on P(S) is defined by dW(µ(dy), ν(dy)) := sup

f∈Lip(S), |f|Lip≤1

• S

f(y)µ(dy) −

• S

f(y)ν(dy)

• .

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) ⊆ Cb(S). Wasserstein metric of order one on P(S) is defined by dW(µ(dy), ν(dy)) := sup

f∈Lip(S), |f|Lip≤1

• S

f(y)µ(dy) −

• S

f(y)ν(dy)

• .

Recall, dW ≤ dTV, and dW metrizes the weak convergence of probability measures. More precisely, {µn}n∈N ⊆ P(S) converges to µ ∈ P(S) with respect to dW if, and only if, lim

n→∞

• S

f(y)µn(dy) =

• S

f(y)µ(dy), f ∈ Cb(S).

Nikola Sandri´ c Probability and Analysis May 15, 2017 10 / 19

Main results Theorem Assume that there is π ∈ P(S) satisfying lim

t→∞ sup s∈T

• S

dW(pt(y, dz), π(dz))ps(x, dy) = 0, x ∈ S. (1) Then, for any p ≥ 1, f ∈ Lip(S) and µ ∈ P(S), 1 t

• [0,t)

f(Ms)τ(ds)

Lp(Ω,Pµ)

− − − − − →

tր∞

• S

f(y)π(dy), (2) where

Lp(Ω,Pµ)

− − − − − →

tր∞

denotes the convergence in Lp(Ω, Pµ).

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 ∈ Cc(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 t

• [0,t)

f(Ms)τ(ds)

Lp(Ω,Pπ)

− − − − − →

tր∞

• S

f(y)π(dy), p ≥ 1, f ∈ Lp(S, π).

Nikola Sandri´ c Probability and Analysis May 15, 2017 12 / 19

Main results As in the strong ergodicity case, the relation in (1) implies that (i) π(dy) is the only measure satisfying (1); (ii) if π(dy) is invariant for M, then it is its unique invariant measure.

Nikola Sandri´ c Probability and Analysis May 15, 2017 13 / 19

Main results As in the strong ergodicity case, the relation in (1) implies that (i) π(dy) is the only measure satisfying (1); (ii) if π(dy) is invariant for M, then it is its unique invariant measure. Question: Does (1) imply invariance of π(dy)?

Nikola Sandri´ c Probability and Analysis May 15, 2017 13 / 19

Main results As in the strong ergodicity case, the relation in (1) implies that (i) π(dy) is the only measure satisfying (1); (ii) if π(dy) is invariant for M, then it is its unique invariant measure. Question: Does (1) imply invariance of π(dy)? If M is a Feller process the answer is yes. Another condition ensuring that (1) implies invariance of π(dy) is contractivity of M with respect to dW: for all t ∈ T and µ, ν ∈ P(S), dW

• S

pt(x, dy)µ(dx),

• S

pt(x, dy)ν(dx)

• ≤ dW(µ(dy), ν(dy)).

Nikola Sandri´ c Probability and Analysis May 15, 2017 13 / 19

Main results Observe that for any t ∈ T and x ∈ S, dW(pt(x, dy), π(dy)) ≤ sup

s∈T

• S

dW(pt(y, dz), π(dz))ps(x, dy).

Nikola Sandri´ c Probability and Analysis May 15, 2017 14 / 19

Main results Observe that for any t ∈ T and x ∈ S, dW(pt(x, dy), π(dy)) ≤ sup

s∈T

• S

dW(pt(y, dz), π(dz))ps(x, dy). Thus, it is tempting to conclude that (1) might be replaced by lim

t→∞ dW(pt(x, dy), π(dy)) = 0,

x ∈ S. (3)

Nikola Sandri´ c Probability and Analysis May 15, 2017 14 / 19

Main results Observe that for any t ∈ T and x ∈ S, dW(pt(x, dy), π(dy)) ≤ sup

s∈T

• S

dW(pt(y, dz), π(dz))ps(x, dy). Thus, it is tempting to conclude that (1) might be replaced by lim

t→∞ dW(pt(x, dy), π(dy)) = 0,

x ∈ S. (3) Theorem Assume that there is π ∈ P(S) satisfying (3), and for every f ∈ Lip(S) and t ∈ T the function Ff,t(x) :=

• S

f(y)pt(x, dy), x ∈ S, is also in Lip(S) with |Ff,t|Lip ≤ Cf, where the constant Cf depends

• nly on f(x). Then, for any f ∈ Lip(S) and µ ∈ P(S), M satisfies the

relation in (2).

Nikola Sandri´ c Probability and Analysis May 15, 2017 14 / 19

Main result As a direct consequence of the contraction property we conclude that for every f ∈ Lip(S) and t ∈ T, Ff,t ∈ Lip(S) and |Ff,t|Lip ≤ |f|Lip.

Nikola Sandri´ c Probability and Analysis May 15, 2017 15 / 19

Main result As a direct consequence of the contraction property we conclude that for every f ∈ Lip(S) and t ∈ T, Ff,t ∈ Lip(S) and |Ff,t|Lip ≤ |f|Lip. Corollary Assume that M is contractive with respect ot dW, and there is π ∈ P(S) satisfying (3). Then, for any f ∈ Lip(S) and µ ∈ P(S), M satisfies the relation in (2).

Nikola Sandri´ c Probability and Analysis May 15, 2017 15 / 19

Examples Let {Xn}n≥1 be a sequence of i.i.d. random variables satisfying P(Xn = 0) = P(Xn = 1/2) = 1/2. Define Mn+1 := 1 2Mn + Xn+1, n ≥ 0, M0 ∈ [0, 1]. Clearly, M is a Markov process with state space ([0, 1], B([0, 1])) and transition function p(x, dy) := P(X1 + x/2 ∈ dy), x ∈ [0, 1]. Also, it is easy to see that Leb(dy) is invariant for M and M is ergodic with respect to Leb(dy). However, observe that, since Leb(dy) is singular with respect to p(x, dy), M is not strongly ergodic.

Nikola Sandri´ c Probability and Analysis May 15, 2017 16 / 19

Examples Let {Xn}n≥1 be a sequence of i.i.d. random variables satisfying P(Xn = 0) = P(Xn = 1/2) = 1/2. Define Mn+1 := 1 2Mn + Xn+1, n ≥ 0, M0 ∈ [0, 1]. Clearly, M is a Markov process with state space ([0, 1], B([0, 1])) and transition function p(x, dy) := P(X1 + x/2 ∈ dy), x ∈ [0, 1]. Also, it is easy to see that Leb(dy) is invariant for M and M is ergodic with respect to Leb(dy). However, observe that, since Leb(dy) is singular with respect to p(x, dy), M is not strongly ergodic. Let d(x, y) := |x − y|, x, y ∈ [0, 1]. Now, by a straightforward computation we get that for all n ≥ 1 and x1, x2 ∈ [0, 1], dW(pn(x1, dy), pn(x2, dy)) ≤ d(x1, x2) 2n .

Nikola Sandri´ c Probability and Analysis May 15, 2017 16 / 19

Examples Let C := C([−1, 0], R). For t ≥ 0 and a function f(s) defined on [t − 1, t], we write f t(s) := f(s + t), s ∈ [−1, 0]. Consider the following stochastic functional differential equation (the so-called stochastic delay equation) dMt = b(Mt)dt + σ(Mt)dBt, t ≥ 0, M0 ∈ C, (4) where b : C → R and σ : C → R. Now, by taking b(f t) = −f t(0) = −f(t) and σ(f t) = g(f t(−1)) = g(f(t − 1)), where g(u) is bounded, Lipschitz continuous, strictly positive and strictly increasing, Hairer, Mattingly and Scheutzow 2011 have shown that (i) the equation in (4) admits a unique strong solution M := {Mt}t≥0 which is a strong Markov and Feller process with state space (C, B(C));

Nikola Sandri´ c Probability and Analysis May 15, 2017 17 / 19

Examples (ii) M possesses a unique invariant probability measure π(dy) which is singular with respect to the transition function pt(x, dy) of M (hence, M is not strongly ergodic); (iii) there are δ > 0, c > 0 and Borel function C : C − → [0, ∞), such that M is contractive with respect to dW, dW(pt(x, dy), π(dy)) ≤ C(x)e−ct and sup

s≥0

PsC(x) < ∞, t ≥ 0, x ∈ C, where d(x, y) := 1 ∧ |x − y|∞/δ, x, y ∈ C.

Nikola Sandri´ c Probability and Analysis May 15, 2017 18 / 19

T h a n k y o u f o r y o u r a t t e n t i o n !

Nikola Sandri´ c Probability and Analysis May 15, 2017 19 / 19