A generalization of Dobrushin coefficient

Transcription

1 A generalization of Dobrushin coefficient Ü µ ŒÆ.êÆ ÆÆ 202.5

2 . Introduction and main results We generalize the well-known Dobrushin coefficient δ in total variation to weighted total variation δ V, which gives a criterium for the geometric ergodicity of discrete-time Markov chains. Let X = (X n ) n 0 be a discrete-time Markov chain taking values on a measurable space (E,E ). Denote P n (x,a) := P[X n A X 0 = x], x E, A E, the n-step transition kernel. Set P = P be the one-step kernel.

3 . Introduction and main results The geometrical ergodicity is, there is an invariant probability measure π on (E,E ) and a constant ρ [0,) and a function C : E (0, ) such that P n (x, ) π Var C(x)ρ n, for all n > 0,π-a.s.x E, () where µ Var := 2sup A E variation for a signed measure µ. µ(a) = sup{ E fdµ : f } is the total

4 . Introduction and main results Then X is said to be strongly ergodic if there exist constants ρ [0,) and C < such that sup P (x, ) π Var Cρ n, for all n > 0. (2) x E

5 . Introduction and main results From [6, 7], we know that () is equivalent to that there exist constants ρ [0,), C < and a π-a.s. finite function V : E [, ] such that P n (x, ) π V CV (x)ρ n, (3) where µ V := sup{ E fdµ : f V } denotes the V -norm on signed measures.

6 . Introduction and main results Recall that for a kernel K (x,a)(x E,A E ), and for a function f, K (x, ) K V := sup V, x E V (x) f (x) f V := sup x E V (x).

7 . Introduction and main results Dobrushin gave a criterion for strong ergodicity. Let δ(p) = 2 sup P(x, ) P(y, ) Var. x E Then, X is strongly ergodic if and only if there exists n, such that δ(p n ) <. δ(p) is now in term of Dobrushin coefficient.

8 . Introduction and main results Now, we will generalize δ(p) to δ V (P) to give a criteria of geometric ergodicity in () or (3). Definition. For a transition kernel P, and a π-a.s. finite function V : E [, ], we define the generalized Dobrushin coefficient δ V (P): δ V (P) := sup x,y E V (x) + V (y) P(x, ) P(y, ) V. (4)

9 . Introduction and main results Here is the main result. Theorem. A ψ-irreducible and aperiodic Markov chain X = (X n ) n 0 is geometriclly ergodic if and only if X is ergodic with stationary distribution π, and there is a π-a.s. finite function V : E [, ] such that π(v ) < and δ V (P n ) < for n large enough. Note that when V (x), δ V (P) = δ(p) and hence Theorem is reduced to the classical Dobrushin criteria.

10 The key ingredient for the proof of Theorem comes from an excellent observation. By defining a metric on E as in [4], we can transfer δ V (P) to obtain a Wasserstein probability space. Following [4], we introduced a metric on E. Let V : E [, ). For x,y E, d V (x,y) = { 0, x = y; V (x) + V (y), x y.

11 Let ϕ be a function on (E,d V ), the Lipschitz norm of ϕ is ϕ(x) ϕ(y) ϕ Lip(dV ) := sup x y d V (x,y) ϕ(x) ϕ(y) = sup x y V (x) + V (y). For two (probability) measures µ, µ 2 on (E,E ), the Wasserstein metric W dv is defined by W dv (µ, µ 2 ) = sup ϕ: ϕ Lip(dV ) ϕd(µ µ 2 ).

12 Lemma 2. [4] For two (probability) measures µ, µ 2 on (E,E ), µ µ 2 V = W dv (µ, µ 2 ), µ i (V ) <, i =,2. (5) For V (x), then both sides of (5) are reduced to the total variation of µ µ 2. The following properties are essential to proof of Theorem.

13 Proposition 3. Let P and P be probability transition kernels on (E,E ). Then (i) δ V (P P) δ V (P)δ V ( P), (ii) δ V (P) δ V ( P) P P V, (iii) Let R be a transition kernel on (E,E ) such that R(x,E) = 0, for all x E and R V <. Then RP V R V δ V (P).

14 Proof. (i) By definition of δ V (P), we have P(x, ) P(y, ) V δ V (P) ( V (x) + V (y) ), π-a.s. x,y E. It follows from Lemma 2 that W dv (P(x, ),P(y, )) δ V (P) ( V (x) + V (y) ). Hence by definition of W dv, we get that V (x) + V (y) Pϕ(x) Pϕ(y) δ V (P) ϕ Lip(dV ), and Pϕ Lip(dV ) δ V (P) ϕ Lip(dV ). (6)

15 Since ( PP)(x,dz)ϕ(z) = (Pϕ)(z) P(x,dz), we get from (6) that ( ) W dv ( PP)(x, ),( PP)(y, ) = sup (Pϕ)(z) ( P(x,dz) ) P(y,dz) ϕ Lip(dV ) δ V (P) sup φ(z) ( P(x,dz) ) P(y,dz) φ Lip(dV ) = δ V (P)W dv ( P(x, ), P(y, ) ).

16 Therefore by Lemma 2 again, we have δ V ( PP) = sup x,y E = sup x,y E V (x) + V (y) ( PP)(x, ) ( PP)(y, ) V V (x) + V (y) W ( ) d V ( PP)(x, ),( PP)(y, ) δ V (P) sup x,y E V (x) + V (y) W ) d V ( P(x, ), P(y, ) = δ V (P) sup x,y E V (x) + V (y) P(x, ) P(y, ) V = δ V (P)δ V ( P).

17 (ii) Using the fact sup we get x u(x) sup x δ V (P) δ V ( P) = sup x,y E V (x) + V (y) P(x, ) P(y, ) V sup x,y E sup x,y E V (x) + V (y) P(x, ) P(y, ) V V (x) + V (y) sup f V v(x) sup u(x) v(x), x ( Pf (x) Pf (y) ) ( Pf (x) Pf (y) ) sup x,y E V (x) + V (y) sup [ ] Pf (x) Pf (x) + Pf (y) Pf (y) f V

18 P(x, ) sup P(x, ) V + P(y, ) P(y, ) V x,y E V (x) + V (y) P(x, ) sup P(x, ) V x E V (x) = P P V, where in the last inequality, we use the element that u(x) + u(y) v(x) + v(y) max{ u(x) v(x), u(y) }. v(y)

19 (iii) Since R(x,E) = 0, for all x E, we have Jordan-Hahn decomposition R = R + R, hence RP V = sup x E = sup x E = sup x E = sup x E V (x) sup (RP)(x,dy)f (y) f V V (x) sup [(R + P)(x,dy) (R P)(x,dy)]f (y) f V V (x) (R+ P)(x, ) (R P)(x, ) V V (x) W ( d V (R + P)(x, ),(R P)(x, ) ).

20 It follows from (6) and Lemma 2 that, ( W dv (R + P)(x, ),(R P)(x, ) ) = sup ϕ(y) ( (R + P)(x,dy) (R P)(x,dy) ) ϕ: ϕ Lip(dV ) = sup ϕ: ϕ Lip(dV ) δ V (P) sup φ: φ Lip(dV ) (Pϕ)(y)[R + (x,dy) R (x,dy)] = δ V (P)W dv ( R + (x, ),R (x, ) ) = δ V (P) R + (x, ) R (x, ) V. φ(y)[r + (x,dy) R (x,dy)]

21 Finally we can obtain RP V δ V (P)sup x E V (x) R+ (x, ) R (x, ) V = δ V (P) R V.

22 Proof of Theorem. Set Π(x, ) = π, for all x E. As noted in [6], the function V in (4) can be chosen to be such that π(v ) <. Since δ V (Π) = 0, we obtain δ V (P n ) = δ V (P n ) δ V (Π) P n Π V = sup x E V (x) Pn (x, ) π V 0. V (x)+π(v ) Conversely, we have I Π V sup V (x) = + π(v ) < x E and (I Π)(x,E) = 0 for all x E. Then P n Π V = P n ΠP n V = (I Π)P n V I Π V δ V (P n ) 0.

23 3.References W.J. Anderson, Continuous-Time Markov Chains, Springer-Verlag, (99). M.F. Chen, From Markov Chains to Non-equilibrium Particle Systems, World Scientufic, Second Edition, (2004). R.L. Dobrushin, Central limit theorem for non-stationary Markov chains I, II, Theorem Probab. Appl., (956), 63-80, M. Hairer and J.C. Mattingly, Yet another look at Harris ergodic theorem for Markov chains, Seminar on Stochastic Analysis, Random Fields and Applications VI Progress in Probability, 20, Volume 63, Part, I. Kontoyiannis and S.P. Meyn, Geometric ergodicity and the spectral gap of non-reversible Markov chains, Probab. Theory Relat. Fields, (20). S.P. Meyn and R.L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag, (996). G.O. Roberts and J.S. Rosenthal. Geometric ergodicity and hybrid Markov chains. Electron. Comm. Probab., 2, 3õ25 (electronic), 997.

24 Thanks!