A Hilbert Space Central Limit Theorem for Geometrically Ergodic Markov Chains

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1 A Hilbert Space Cetral Limit Theorem for Geometrically Ergodic Marov Chais Joh Stachursi Research School of Ecoomics, Australia Natioal Uiversity Abstract This ote proves a simple but useful cetral limit theorem for Hilbert space valued fuctios of geometrically ergodic Marov chais o geeral state spaces. The theorem is valid for chais startig at a arbitrary poit i the state space. Keywords: Marov chais, Hilbert space, Cetral limit theorem 1. Itroductio Let (X t ) t 1 be a geometrically ergodic Marov chai o state space X (full defiitios follow) ad let π be the uique statioary distributio. It is well-ow (see, for example, 5] or 8, chapter 17]) that if T is a measurable fuctio from the state space X to R satisfyig a suitable secod momet coditio, the 1/2 T(X t) Tdπ] coverge i law to a cetered Gaussia distributio o R. Usig the Cramer- Wold device, the same result ca be exteded without techical difficulties to the case where T taes values i R. 1 I this paper we provide a aalogous CLT result for the case where T taes values i is a separable Hilbert space. The aim is ot to provide a particularly geeral Hilbert cetral limit theorem for depedet variables, but rather to provide a set of coditios that are straightforward to chec i applicatios. The proof of our result is based o the depedet variable Hilbert CLT of Merlevède et al. 7]. 2. Set Up Let (Ω, F, P) deote a arbitrary probability space o which all radom variables are supported. As usual, if (E, B) is ay measurable space, the a E-valued radom variable X is a measurable map from (Ω, F ) to (E, B). We use the symbol LX to deote its law (i.e., LX = P X 1 ). I what follows, if E has a topology the, the σ-algebra B is always tae to be the Borel sets. Uless otherwise stated, measurability of fuctios refers to Borel measurability. If µ is a measure o (E, B) ad h is a real-valued measurable fuctio o E, the µ(h) deotes hdµ wheever the latter is defied. If E is a topological space ad (µ ) 0 are probabilities (i.e., Borel probability measures) o E, the µ µ 0 i distributio if µ (h) µ 0 (h) i R for every cotiuous bouded h : E R. The sequece (µ ) 1 is called tight if for all ε > 0 there is a compact K E with sup 1 µ (E \ K) ε. Below we cosider a stochastic process taig values i a separable Hilbert space H. Let deote the orm o H, ad h, g the ier product of h ad g. If Y is a H-valued radom variable with E Y <, the, by the Riesz represetatio theorem, there exists a uique elemet EY of H such that E h, Y = Fiacial support from Australia Research Coucil grat DP is gratefully acowledged. address: joh.stachursi@au.edu.au (Joh Stachursi) 1 For applicatios of this CLT focusig o Marov chai Mote Carlo, see the surveys of Roberts ad Rosethal 10] ad Joes 5]. For a applicatio i ecoomics see 9]. For other applicatios see 8]. Preprit submitted to Elsevier March 21, 2012

2 h, EY for all h H. The vector EY is called the expectatio (or Pettis itegral) of Y. For ay H-valued radom variable Y with E Y 2 < ad EY = 0, the covariace operator C : H H of Y is defied by g, Ch = E g, Y h, Y for all g, h H. A radom variable V taig values i H is called Gaussia if h, V is Gaussia o R for each h H. To simplify the presetatio, i what follows we regard degeerate radom variables o R as Gaussias with zero variace Mai Result Let (X, X ) be a measure space, ad let P be a stochastic erel o X. I particular, P(x, dy) is a probability measure o (X, X ) for each x X, ad x P(x, B) is measurable for every B X. I what follows, we use the stadard otatio (ψp)(b) := P(x, B)ψ(dx) ad (P f )(x) := f (y)p(x, dy). Here ψ is a probability measure o (X, X ) ad f : X R is a measurable fuctio such that the itegral is defied. Let P t deote the t-th iterate of either oe of these operators. A probability π o (X, X ) is called statioary for P if πp = π. Let TV be the total variatio orm over the space of fiite siged measures o (X, X ). We assume throughout that P is geometrically ergodic, which is to say that (i) P has a uique statioary distributio π, (ii) ψp t ϕp t TV 0 as t for ay probabilities ψ ad ϕ o (X, X ), ad (iii) there exists a measurable fuctio V : X 0, ) ad costats R R + ad α 0, 1) such that Vdπ < ad sup P t (x, B) π(b) α t RV(x) for all x X, t N (1) B X Sufficiet coditios for geometric ergodicity are discussed i may sources. See, for example, 8] ad 4]. See also 6, Theorem 21.12] for a rage of coditios equivalet to (ii). Lettig ψ be a probability measure o X, we call a X-valued stochastic process (X t ) t 1 Marov-(P, ψ) if X 1 is draw from ψ ad P is the trasitio probability fuctio for (X t ) t 1. More formally, this meas that Eh(X t+ ) F t ] = P h(x t ) (2) almost surely for ay t, N ad ay bouded measurable h : X R, ad, i additio, LX 1 = ψ. Here F t is the σ-algebra geerated by (X 1,..., X t ), ad E F t ] is coditioal expectatio with respect to F t. Existece of at least oe such a sequece (X t ) t 1 follows from a well-ow theorem of Ioescu-Tulcea (see, e.g., 11, theorem II.9.2]). If ψ is a Dirac probability measure cocetrated at a sigle poit x, the we call (X t ) t 1 Marov-(P, x). If (X t ) t 1 is Marov-(P, π), the (X t ) t 1 is statioary, ad LX t = π for all t (see, e.g., 8, chapter 3]). Our mai result cocers sequeces of the form T 0 (X t )] t 1, where T 0 is a measurable map from X ito a separable Hilbert space H. O T 0 we impose the followig assumptio: Assumptio 3.1. There exists oegative costats m 0, m 1 ad γ < 1 such that T 0 (x) 2 m 0 + m 1 V(x) γ for all x X. The followig lemma assures us that if LX = π, the E T 0 (X) exists. Lemma 3.1. If LX = π ad assumptio 3.1 holds, the E T 0 (X) <. 2 For more details o Hilbert-space valued stochastic processes, see, for example, 1]. 2

3 Proof. Assume the coditios of the lemma. It suffices to show that E T 0 (X) 2 <. Applyig assumptio 3.1 ad Jese s iequality, we have E T 0 (X) 2 m 0 + m 1 EV(X) γ ] m 0 + m 1 EV(X)] γ The fial expressio is fiite by the left-had side of (1). We eed two fial defiitios. Let (X t ) t 1 be Marov-(P, π). By lemma 3.1, E T 0 (X 1 ) exists i H. Defie T : X H be the map T(x) = T 0 (x) E T 0 (X 1 ) (x X), ad let C be the covariace operator defied by g, Ch = E g, T(X 1 ) h, T(X 1 ) + E g, T(X 1 ) h, T(X t ) + E h, T(X 1 ) g, T(X t ). (3) t 2 t 2 for g, h H. We ca ow state our mai result: Theorem 3.1. Let assumptio 3.1 hold. If x X ad (X t ) t 1 is Marov-(P, x), the ] L 1/2 T(X t ) N(0, C) ( ). (4) Here N(0, C) represets the distributio of a H-valued Gaussia radom variable with expectatio equal to the origi of H ad covariace operator C Example Before turig to the proof of theorem 3.1, we preset a simple illustratio. Let µ be ay probability measure o (R, B), ad cosider the separable Hilbert space L 2 := L 2 (R, B, µ). Let P be a geometrically ergodic stochastic erel o R, ad let F be the cumulative distributio fuctio of its statioary distributio. I may cases, o closed form expressio for F is available. Suppose that we wish to compute it by simulatio. A atural techique is to pic ay x R, simulate a Marov-(P, x) process (X t ) t 1, ad evaluate the empirical cumulative distributio fuctio F (y) := 1 1{X t y. Let us ivestigate the error F F, measured i L 2 orm. Defie T 0 (x) to be the fuctio y 1{x y. We the have T 0 (x) 2 = 1{x y 2 µ(dy) = µ(x, )) 1. Taig m 0 = 1 ad m 1 = 0, we see that assumptio 3.1 is alway satisfied. Moreover, a straightforward applicatio of Fubii s theorem shows that if LX 1 = F, the E T 0 (X 1 ) = F. As a result, settig T := T 0 F, theorem 3.1 gives (F F) = { 1 T 0 (X t ) F = 1/2 T(X t ) N(0, C) where C is defied by (3). As a corollary, cotiuity of the orm ow implies that F F = O P ( 1/2 ). 4. Proof of theorem 3.1 Our first lemma shows that, give our ergodicity assumptios o P, we ca restrict attetio to the case where LX 1 = π whe provig (4). Lemma 4.1. Let (X t ) t 1 ad (X t ) t 1 be two P-Marov chais, where LX 1 = π ad X 1 = x X. For ay Borel probability measure ν o L 2 (µ), ] ] L 1/2 T(X t ) ν implies L 1/2 T(X t) ν 3

4 Proof. Give our assumptio of geometric ergodicity (ad hece ergodicity), it is well ow (see Lidvall, 6, Theorem 21.12]) that oe ca costruct P-Marov processes (X t ) t 1 ad (X t ) t 1 o a commo probability space (Ω, F, P) such that τ := if{t N : X t = X t is fiite almost surely, ad X t = X t for all t τ. Let S := T(X t) ad S := T(X t ), ad assume as i the statemet of the lemma that 1/2 S ν. To prove that 1/2 S ν it suffices to show that the (orm) distace betwee 1/2 S ad 1/2 S coverges to zero i probability (cf., e.g., Dudley, 3, Lemma ]). Fixig ε > 0, we eed to show that P{ 1/2 S 1/2 S > ε 0 ( ) (5) Clearly { 1/2 S 1/2 S > ε { T(X t) T(X t ) > 1/2 ε Fix N, ad partitio the last set over {τ ad {τ > to obtai the disjoit sets ad { T(X t) T(X t ) > 1/2 ε { T(X t) T(X t ) > 1/2 ε { {τ T(X t) T(X t ) > 1/2 ε {τ > {τ > Together, these lead to the boud { { 1/2 S 1/2 S > ε T(X t) T(X t ) > 1/2 ε {τ > { P{ 1/2 S 1/2 S > ε P T(X t) T(X t ) > 1/2 ε + P{τ > For ay fixed, we have Hece Sice P{τ < = 1 taig yields (5). { lim P T(X t) T(X t ) > 1/2 ε = 0 (6) lim sup P{ 1/2 S 1/2 S > ε P{τ >, N I view of Lemma 4.1, we ca cotiue the proof of (4) while cosiderig oly the case LX 1 = π. I this case (T(X t )) is a cetered strict sese statioary stochastic processes i H, ad we ca apply the statioary Hilbert CLT i Merlevède et al. 7, Theorem 4, Corollary 1]. From the latter we obtai the followig result: Let ξ t := T(X t ) for all t. Defie the correspodig mixig coefficiets by α(t) := sup P(A B) P(A)P(B) where the supremum is over all A σ(ξ 1 ) ad B σ(ξ t+1 ). I this settig, the covergece i (4) will be valid wheever there exists a costat δ > 0 such that E ξ t 2+δ < ad 4 t 2/δ α(t) < (7)

5 (The defiitio of the mixig coefficiet used here is slightly differet to the oe used i Merlevède et al. 7, Defiitio 1]. However, i the Marov case it is well-ow that the two are equivalet. See, for example, Bradley 2, Sectio 3].) We establish first the fiite expectatio o the left-had side of (7). Let m 0, m 1, γ ad V be the costats ad fuctio i assumptio 3.1. Let r := E T(X 1 ) 2. Evidetly T(x) 2/γ = T 0 (x) E T(X 1 ) 2/γ From this boud, assumptio 3.1 ad Jese s iequality, we obtai T(x) 2/γ 2m 0 + 2m 1 V(x) γ + 2r] 1/γ 1 3 I other words, there exist fiite costats c 1 ad c 2 such that ] 1/γ 2 T 0 (x) 2 + 2r ξ t 2/γ := T(X t ) 2/γ c 1 V(X t ) + c 2 {6m 0 ] 1/γ + 6m 1 V(x) γ ] 1/γ + 6r] 1/γ holds poitwise o Ω. Let δ := 2(1 γ)/γ, so that 2/γ = 2 + δ. Taig expectatios ad applyig the first expressio i (1) gives E ξ t 2+δ < as required. The last step of the proof of Theorem 3.1 is to verify the fiiteess of the sum o the right-had side of (7). A elemetary argumet shows the followig orderig of σ-algebras: σ(ξ j ) = σ(t(x j )) σ(x j ), j As a result, we have α(t) := sup P(A B) P(A)P(B) sup P(A B) P(A)P(B) A σ(ξ 1 ) A σ(x 1 ) B σ(ξ t+1 ) B σ(x t+1 ) The right-had side gives the strog mixig coefficiets for (X t ), which, i the geometrically ergodic case, are ow to be O(λ t ) for the costat λ i (1). (See, for example, Joes 5, p. 304].) As a cosequece, we have α(t) = O(λ t ), ad hece t2/δ α(t) will be fiite if t2/δ λ t is fiite. Sice λ < 1, this last sum is clearly fiite. This completes the proof of Theorem 3.1. Refereces 1] Bosq, D., Liear Processes i Fuctio Space, Spriger-Verlag. 2] Bradley, R. C. (2005): Basic properties of strog mixig coditios: A survey ad some ope questios, Probability Surveys, 2, ] Dudley, Richard M. (2002): Real Aalysis ad Probability, Cambridge Studies i Advaced Mathematics No. 74, Cambridge Uiversity Press. 4] Hairer, M. ad J. C. Mattigly (2011): Yet aother loo at Harris ergodic theorem for Marov chais, i Semiar o Stochastic Aalysis, Radom Fields ad Applicatios VI (R. C. Dalag, M. Dozzi ad F. Russo, ed.) Spriger Basel. 5] Joes, G.L., O the Marov chai cetral limit theorem, Probab. Surv., 1, ] Lidvall, T. (2002): Lectures o the Couplig Method, Dover Publicatios, Mieola N.Y. 7] Merlevède, F., M. Peligrad ad S. Utev (1997): Sharp coditios for the CLT of Liear Processes i a Hilbert Space, Joural of Theoretical Probability, 10 (3),

6 8] S. Mey, Tweedie, R.L., Marov Chais ad Stochastic Stability, 2d Editio, Cambridge Uiversity Press, Cambridge. 9] Nishimura, K., Stachursi, J., Stability of stochastic optimal growth models: A New Approach, J. Eco. Theory, 122 (1), ] Roberts, G.O., Rosethal, J.S., Geeral state Marov chais ad MCMC algorithms, Probab. Surv., 1, ] Shiryaev, A.N., Probability, Spriger-Verlag, New Yor. 6