Three Kinds of Geometric Convergence for Markov Chains and the Spectral Gap Property

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1 E l e c t r o n c J o u r n a l o f P r o b a b l t y Vol. 6 (0), Paper no. 34, pages Journal URL Three Knds of Geometrc Convergence for Markov Chans and the Spectral Gap Property Wolfgang Stadje and Achm Wübker Insttute of Mathematcs, Unversty of Osnabrück AlbrechtstraSSe 8a, Osnabrück, Germany E-mal: wolfgang@mathematk.un-osnabrueck.de E-mal: awuebker@mathematk.un-osnabrueck.de Abstract In ths paper we nvestgate three types of convergence for geometrcally ergodc Markov chans (MCs) wth countable state space, whch n general lead to dfferent rates of convergence. For reversble Markov chans t s shown that these rates concde. For general MCs we show some connectons between ther rates and those of the assocated reversed MCs. Moreover, we study the relatons between these rates and a certan famly of sopermetrc constants. Ths sheds new lght on the connecton of geometrc ergodcty and the so-called spectral gap property, n partcular for non-reversble MCs, and makes t possble to derve sharp upper and lower bounds for the spectral radus of certan non-reversble chans. Key words: Markov chan, countable state space; geometrc ergodcty; spectral gap property; sopermetrc constant; reversblty; bounds for the spectral radus. AMS 00 Subject Classfcaton: Prmary 60J0. Submtted to EJP on October 9, 00, fnal verson accepted Aprl 0, 0. Research has been supported by the DFG 00

2 Introducton For postve recurrent Markov chans (MCs) one of the central questons s the convergence of ther transton kernels to the nvarant dstrbuton. The geometrcally ergodc case when ths convergence takes place at a geometrc rate s of partcular mportance. A profound analyss of ths subject can be found n the monographs by Meyn and Tweede [7] and by Nummeln [8]. In ths paper we are concerned wth three dfferent knds of rates of geometrc convergence. In Secton we present an example to llustrate the dfferences between the defntons; n Secton 3 several connectons between these rates for a MC and the correspondng rates for the reversed chan are proved. In Secton 4 we show that for reversble Markov chans (under a mld condton) the dfferent types of rates of convergence actually concde. In Secton 5 we analyze geometrcally ergodc MCs by applyng the concept of sopermetrc constants, whch has been used n [4] to establsh necessary and suffcent condtons for the spectral gap property. We show that ths property and geometrc ergodcty are equvalent for normal Markov chans, generalzng the results of Roberts and Tweede [] and Roberts and Rosenthal []. Moreover, t s shown how a certan sequence of sopermetrc constants can be used to obtan bounds for the rates of geometrc convergence, and prove that these bounds are sharp n some cases. In Secton 6 we present an example whch shows that geometrc ergodcty (GE) does not mply the spectral gap property (SGP) and calculate exact rates of geometrc convergence applyng the method of sopermetrc constants. Throughout ths paper let ξ, ξ,... be a postve recurrent MC wth countable state space Ω, transton kernel p(, ) and nvarant probablty measure π. Let p π(j)p(j, ) (, j) =,, j Ω () π() be the transton probabltes of the reversed MC (a realzaton of whch we denote by ξ, ξ,...). We need the standard MC operators P, P and Π defned by P f () = f (j)p(, j), () j Ω P f () = j Ω f (j)p (, j), (3) Π f () = f (j)π(j). (4) j Ω for all real-valued functons f on Ω for whch the correspondng seres converge. In partcular, for all f L (π) t easly follows from Jensen s nequalty and the statonarty of π that the sums n (), (3) and (4) converge and that P f, P f and Π f are n L (π). Note that we consder Π as the operator that maps every f L (π) to the functon constantly equal to the π-expected value of f. The scalar product on L (π) s of course f, g π = f (j)g(j)π(j). j Ω It s easy to show that P f, g π = f, P g π, 00

3 so P s the adjont operator of P on L (π). We say that P has the spectral gap property (SGP) on L (π) f where and ρ = lm sup P n f f L0, (π) n L (π) L 0, (π) = { f L (π) : f L (π) = 0, f L (π) = }, f L (π) = j Ω f (j)π(j), f L (π) = <, (5) j Ω f (j) π(j) /. Note that the lmt n (5) always exsts (see e.g. [0]). The total varaton dstance of two probablty measures µ and ν on Ω s defned by d(µ, ν) = µ ν T V = sup (µ(j) ν(j))φ(j). If we set A = { j Ω : µ(j) ν(j)}, then clearly φ: φ = j Ω d(µ, ν) = µ(a) ν(a). A Markov chan ξ, ξ,... s called geometrcally ergodc (GE) f for some δ < p n (, ) π T V K δ () = sup n N δ n < Ω. (6) From [7] (Chapter 5) and [8] (Theorem 6.4 ()) t follows that the GE property s equvalent to the seemngly more restrctve condton K δ L (π) = K δ ()π() < (7) = for some δ <, where K δ s defned as n (6). Note that the δ n (7) may dffer from the δ n (6). Obvously, (7) mples that for some δ < C(δ) = sup n N Ω pn (, ) π T V π() δ n <. (8) It s certanly of nterest to fnd the best rate of geometrc convergence. However, consderng (6)-(8) there are three possbltes to defne an optmal lower bound for ths rate: Let δ 0 = nf{δ : 0 < δ < and (6) s satsfed} (9) δ = nf{δ : 0 < δ < and (7) s satsfed} (0) δ = nf{δ : 0 < δ < and (8) s satsfed}. () Defnton. Regardng the geometrc rate of convergence we call δ 0 the optmal lower bound (OLB) n the weak sense, δ the OLB n the strong sense and δ the OLB n the L (π) sense. It follows from the defntons that δ δ δ 0. Are these nequaltes n general strct, and under whch condtons do they become equaltes? Moreover, are these OLBs attaned? We start wth an example. 003

4 Introductory example: the reversed wnnng streak Let us consder the MC wth state space N and transton matrx () Its nvarant measure π s gven by The crucal observaton now s that whch mmedately generalzes to It follows that For arbtrary δ > 0 we conclude that π() =, N. p(, ) = π() N, p (, ) π T V = 0 N. p j (, ) π T V = 0 j, N. p n (, ) π T V (/δ) δ n n N, N. Snce ths holds true for all δ > 0, we see that K δ () (/δ) and that the OLB n the weak sense s zero,.e., δ 0 = 0. But of course the MC s not GE at rate zero (ths rate of geometrc ergodcty only occurs for MCs nduced by a sequence of..d. random varables); thus the nfmum n (9) s not attaned. Next let us determne δ. Check that p ( +, ) π() = N. Now consder an arbtrary δ < satsfyng (0). Then so that p ( +, ) π T V K δ ( + )δ, K δ () δ/ δ N. 004

5 So (7) holds for δ only f δ <, δ = whch s of course equvalent to δ > /. Hence, δ. (3) On the other hand, f we choose δ = +ε, we see that for any ε (0, ) we have that K δ() ( ε). Moreover, a smple calculaton shows that (7) s satsfed. Ths together wth (3) mples that δ =. The above reasonng mples that ths MC s not GE wth rate n the strong sense. Regardng δ, so far we only know that δ. Its exact value wll be derved n the next secton, where we wll also see how the dfferent rates of convergence occur n a natural way when tryng to bound δ 0, the OLB of the reversed chan n the weak sense, by the OLBs of the orgnal MC. 3 The reversed chan Assumng that a MC ξ, ξ,... s GE, what can we say about the reversed MC ξ, ξ,...? We show that the GE property s preserved under tme-reverson, but the behavor of the OLBs s more complcated. Theorem. If a MC s GE, then the reversed MC s also GE. Proof: Let B (n) = { j Ω : p n (, j) π(j)} and δ (δ, ). Then we have and C(δ) < snce δ > δ. Actually, we have just shown p n (, ) π T V = p n (, B (n) ) π(b (n) ) π(j) = p n (j, ) π() π() j B (n) π() π() j B (n) π(j) pn (j, ) π T V δ n j Ω π(j) pn (j, ) π T V δ n = C(δ) π() δn (4) δ n δ n 005

6 Corollary. If ξ, ξ,... s GE, then δ 0 δ. Theorem. If ξ, ξ,... s GE, then where δ denotes the OLB of the reversed MC n the L (π) sense. δ = δ, (5) Proof: We have p n (, ) π T V π() = p n (, B (n) ) π(b (n) ) π() Ω Ω = = Ω j B (n) Ω j B (n) π(j) p n (j, ) π() π() π() π(j) p n (j, ) π() π(j) p n (j, ) π() j Ω Ω = p n (, ) π T V π(). (6) Ω For every δ > δ there s a constant C such that the rght-hand sde of (6) s at most Cδ n for all n. It follows that δ δ. Usng the fact that p (, ) = p(, ) and carryng out the same calculatons as n (6) wth p (, ) nstead of p (, ), we obtan δ δ. Let us apply Theorem to the example n Secton. The transton matrx of the reversed MC s gven by (7) Ths MC has a remarkable feature: there s a central state n the sense that ths state can be reached from any other one n a sngle step wth probablty /. Ths property mmedately mples that sup p (, A) p (j, A), j Ω,A Ω. (8) It s nterestng that (8) mples the classcal condton whch was used by Döbln [] n order to establsh unform geometrc convergence to the nvarant measure (wth respect to total-varaton) for certan Markov chans,.e., p n (, ) π T V δ < : sup sup n Ω δ n Note that ths s a stronger property than (6). = sup K δ () <. Ω 006

7 In [6] t s shown that (8) mples that n p n (, ) π T V (9) (the constant does not appear n [6] due to a dfferent defnton of the total varaton norm). The proof s based on a couplng argument n whch (8) s used to bound the expected couplng tme, whch n turn leads to the estmate for the total varaton (see [6]). The factor n (9) s optmal n the sense that t s as small as possble. In fact, sup p n (, ) π T V p n (, n) π(n) = n, Ω so δ 0 =. From (9) t now follows mmedately that δ 0 = δ = δ =. (0) The stuaton s completely dfferent from what we have seen for the orgnal chan, for whch t has been shown that 0 = δ 0 δ δ =. Let us determne δ, whch had been left open at the end of Secton. From Theorem and (0) t follows that δ = δ =. A closer look at the proof of Theorem yelds even more. We obtan n p n (, ) π T V π() 4, Ω so the OLB n the L (π) sense, δ, s n fact attaned. Recall that ths was not the case for δ 0 and δ. 4 Reversble Markov chans In ths secton we show that for reversble MCs δ 0, δ and δ concde under the (rather weak) condton that the nvarant dstrbuton π has a fnte (+ε)-moment (ε > 0),.e., f M = = +ε π() <. Theorem 3. If a MC s reversble, GE and ts nvarant dstrbuton π has a fnte ( + ε)-moment for some ε > 0, then δ 0 = δ = δ, () and all these OLBs are attaned. Proof: Wthout loss of generalzaton we can assume that Ω = N and π() π( + ) N. 007

8 Defne δ : N R by and δ (k) = : = k 0 : k ρ = lm sup (P n Π) δ π n, () L (π) wth f L (π) = [ j Ω f (j) π(j)]] /. Now we apply the spectral representaton theorem (see e.g. [0]) wth spectral measure δ /π δ /π ν (λ) = E λ, δ /π L (π) δ /π L (π) assocated to P Π and (δ /π)/ δ /π L (π), E λ denotng the correspondng projecton operator. We obtan (P n Π) δ π n From () and (3) t follows that L (π) = = (P n Π) δ = δ π n = π, δ π n L (π) λ n δ d E λ π, δ π L (π) L (π) λ n ν (dλ) n n n λ n ν (dλ) π() n (3) ρ = max[ nf supp(ν (λ)), sup supp(ν (λ))]. (4) We have p n (, ) π T V = sup p n (, j) π(j) φ(j) φ: φ = j Ω δ (k) = sup p n (k, j) π(j) φ(j)π(k) φ: φ = π(k) j Ω k Ω δ = sup φ: φ = π, (P n Π)φ = sup (P n Π) δ φ: φ = π, φ L (π) L (π) (P n Π) δ π L (π) ρ n π() (5) 008

9 sup ρ n j j Ω π() (6) ρ n (7) π() where the frst two nequaltes follow from Cauchy-Schwarz and the denttes (3)-(4), respectvely. The last nequalty follows from the defnton of ρ. From the equvalence of () and () n Theorem. of [] t follows that the upper bound ρ for the rate n (5) s optmal n the sense that p n (, ) π T V sup n δ n = δ < ρ Ω. Ths mples that δ 0 = sup ρ j. j Ω From (6) t follows that δ 0 s attaned,.e., that (6) holds for δ = δ 0. Now let us prove (). By (6), t s enough to show that Let K = = (+ε). We obtan L (π) = π() <. (8) π( ) = π() = From the last proof we mmedately obtan = = +ε π() +ε K M <. (9) Corollary. For a reversble MC the followng two statements are equvalent:. ρ = sup j Ω ρ j.. δ 0 = ρ. The estmate n (7) s the well-known π() -bound for the total varaton n terms of the spectral radus. For Markov chans wth fnte state space ths can be found n [3]. 5 Geometrc ergodcty and spectral theory The followng theorem due to [] and [] shows the close connecton between geometrc ergodcty and the spectral gap property. Theorem 4. For a reversble MC ξ, ξ,... the followng two statements are equvalent: 009

10 . ξ, ξ,... s GE.. P satsfes the SGP. Moreover, ρ = δ 0. The orgnal proof of ths result can be found n []. A very short dervaton of the frst part was gven n [4]. The key observaton there was that the spectral radus of a MC can be expressed by a rescaled functon of a sequence of sopermetrc constants (see Theorem 5 below). It turns out that these rescaled constants are a sutable tool for studyng geometrc ergodcty n the sense that they can be related to the dfferent notons of geometrc speed of convergence. The sopermetrc constants n queston are k n = nf A Ω k n(a), k P n P n = nf A Ω k P n P n(a), n N where k n (A) = π(a)π(a c p n (, A c )π() ) A k P n P n(a) = π(a)π(a c p n (, j)p n (j, A c )π(). ) A j Ω The followng theorem from [4] relates spectral propertes to the rescaled lmts of sopermetrc constants. Theorem 5. Assume that the operator P s normal. Then the spectral radus ρ s gven by In partcular, for reversble Markov chans ths yelds Moreover, f P s n addton postve, we have Based on ths result, we can show Theorem 6. If the underlyng MC s GE, then sup A Ω ρ = lm kp n n P n. (30) n ρ = lm k n. ρ = lm kn n. lm sup( k P n P n(a)) n δ. If P s n addton normal, then the MC satsfes SGP and the spectral radus ρ can be estmated by δ 0 ρ δ. (3) 00

11 Proof: An easy calculaton shows that Hence, for every ε (0, δ ), k P n P n(a) = π(a)π(a c ) lm sup( k P n P n(a)) n = lm sup ε + δ lm sup (p n (, A c ) π(a c )) π(). (3) Ω π(a)π(a c (p n (, A c ) π(a c )) π() ) Ω n n Ω pn (, ) π T V π() (ε + δ ) n π(a)π(a c ) ε + δ. (33) Ths proves the frst asserton of the theorem. n The frst nequalty n (3) follows from the second part of Theorem 4. Let us prove the second nequalty. It was shown n [4] that for l < n we have Thus, by (34) and (3), ( k P l P l (A)) l ( k P l P l (A)) l ( k P n P n(a)) n. (34) π(a)π(a c (p n (, A c ) π(a c )) π() ) π(a)π(a c ) Ω n n p n (, ) π T V π() Ω n. (35) Now frst lettng n, then takng the supremum over all A Ω, thereafter lettng l and applyng Theorem 5 yelds ρ δ. From ths theorem we mmedately obtan Corollary 3. If P s normal, then the followng statements are equvalent:. ξ, ξ,... s GE.. ξ, ξ,... satsfes SGP. Next we want to prove the equvalence n Corollary 3 for certan non-reversble MCs. Note that normalty of the operator P s only needed to ensure that (34) holds. So t seems natural to start wth a modfed verson of (34). Defne a(n, A) = ( k P n P n(a)) n. (36) 0

12 Corollary 4. Assume that for every A Ω the sequence (a(n, A)) n N has a nondecreasng subsequence (a(n k, A)) k N wth n =. Then the GE property and SGP are equvalent and ρ where κ s a constant whch does not depend on the underlyng MC. κ 8 δ, (37) Note that the subsequence (n k ) k s allowed to depend on A. The fact that κ has been establshed n [5], from whch the followng defnton of κ s taken: Let denote the set of all possble dstrbutons of pars (X, Y ) of..d random varables each havng varance. Then κ = nf sup E (X + c) (Y + c) c R E((X + c). (38) ) Proof: The mplcaton SGP = GE can be derved n a smlar way as (5). More precsely, n the dervaton of (5) we have to take the adjont n the nner product,.e. to replace P n Π by P n Π. The result follows by applyng Cauchy-Schwarz n (5) and the fact that P n Π L (π) = P n Π L (π). GE = SGP follows mmedately from (37), snce δ < mples ρ <. So let us show (37). Snce (a(n k, A)) k N s nondecreasng, we can carry out the same calculaton as n the proof of Theorem 6 wth n replaced by n k. By assumpton, we have n = for all A Ω. Ths yelds whch mples that Now (37) follows from Proposton of [6]. ( k P P(A)) δ, (39) ( k P P) δ. Because of ts generalty, the upper bound n (37) s not sharp n most cases. In order to mprove ths upper bound for certan MCs we show the followng generalzaton of Theorem 5. We need the Hlbert space L 0 (π) = { f L (π) : j Ω f (j)π(j) = 0}. Theorem 7. For a postve recurrent MC the spectral radus ρ = ρ(p) of the assocated Markov operator P on L 0 (π) s gven by ρ = lm lm k(p n P l n ) l n l. (40) Proof: Snce P n P n s postve and selfadjont, Theorem 5 yelds ρ(p n P n ) = lm k(p n P n ) l l. l By the Raylegh-Rtz prncple (see e.g. [5]) t follows that sup P n P n f, f π = lm k(p n f L0, (π) P n ) l l. (4) l 0

13 Snce the left-hand sde n (4) equals P n L0 (π), we obtan Now n leads to the asserton. P n n = lm k(p L0 (π) n P l n ) l n l. Corollary 5. Assume that there exsts an n 0 N such that P n P n = (P P) n n n 0. (4) Then and ρ(p) = ρ(p P) = lm kp n P n n δ 0 ρ(p) δ. (43) Proof: From Theorem 7 t follows that ρ(p) = lm lm k(p n P n ) l n l l = lm lm k(p P) n l n l l = ρ(p P) = lm k(p P) n n = lm k(p n P n ) n. (44) The nequaltes (43) can be shown n the same way as n the proof of Theorem 6. The upper bound n (43) s better than that n (37). To show ths, note that snce we do not know the exact value of κ, the estmate (37) can only be appled wth κ =. Therefore we have to prove that δ δ, 8 whch s equvalent to 8 ( δ ) ( + δ ) δ. Actually, δ s smaller than the rght-hand sde of (37) whenever max δ [0,] ( δ)( + δ) 8/κ. Ths s the case as long as κ 7/4. Observe that normalty of a MC mples condton (4). Let us agan consder the example of Secton to show that ths mplcaton cannot be reversed. Let P and P be gven by () and (7), respectvely. It can be readly seen that for and j N we have (P P),j = π j + δ,j 03

14 and (P P ), j = δ 0, j + δ,j. Ths mples that P P P P, so the MC s not normal. However, a short calculaton shows that By (45), ((P P) ), j = 3 4 π j + 4 δ, j = (P P ),j. (45) P 3 P 3 = P (P P )P = P (P P) P = P P P P = P P (P P )P P = (P P) (P P) (P P) = (P P) 3. (46) By complete nducton, t s now seen that (4) s satsfed wth n 0 =. The spectral gap n ths example has already been determned n [4]. We gve a very short alternatve dervaton. From Corollary 5 t follows that ρ(p) = ρ(p P). But P P = I + Π, (47) where I denotes the dentty operator,.e., I f = f. Snce P P s selfadjont, we obtan ρ(p) = Note that the nequalty ρ δ = = = ρ(p P) = P P L 0 (π) We can use ths n order to obtan an estmate for κ. Insert ρ = I + Π L0 (π). (48), whch has been derved n Corollary 5, s n fact sharp! nto (37) we obtan that κ The computatons n the proof of Theorem 3 lead to the followng modfcaton of Corollary 5: Corollary 6. If the operator P of a geometrcally ergodc MC satsfes (4) and the nvarant dstrbuton π has a fnte ( + ε)-moment for some ε > 0, then δ ρ δ. The followng result provdes lower bounds for δ 0 and δ. 04

15 Theorem 8. If the MC s GE, δ 0 δ sup lm sup k n (A) n. (49) A Ω sup A Ω:mn( A, A c )< lm sup k n (A) n. (50) If for every A Ω the sequence ( k n (A) n ) n N s nondecreasng, we even have Moreover, for every sequence (A n ) n N wth lm δ lm k n n. (5) π(a n )π(a c n ) n = we have δ lm sup k n (A n ) n. (5) Proof: We only show the thrd nequalty of Theorem 8 because the proofs of the others are smlar. We have by assumpton that, for arbtrary δ > δ, k n0 (A) n 0 lm k n (A) n = lm π(a)π(a c ) lm lm sup p n (, A c )π() A n lm sup π(a)π(a c p n (, ) π T V π() ) A n p n (, ) π T V π() Ω Now δ δ and n 0 yelds the result. n n lm sup C(δ) n δ = δ. (53) Let us apply ths result to our example. A good choce of the set A s of key mportance n order to obtan a non-trval lower bound. We try A = {, 4, 6, 8,...}. Then k n (A) = = π(a)π(a c p n (, A c )π() ) A n π(a)π(a c π(a c /4 (/4)n+ n )π() = 3 =. ) 3/4 4 = (54) Ths mples that ( k n (A)) n = 05

16 for all n. Applyng Theorem 8 yelds δ. By what has been shown before, ths bound s agan sharp. One can prove that the above choce of A s optmal n the sense that k n (A) = k n. So we have just seen that n our example we have ( k n ) n = δ n. (55) It would be nce to have ths relaton n general, at least asymptotcally, but ths result fals to be true. In the next secton we consder an example (orgnally due to Häggström [3]) of a MC that s GE and satsfes k n = 0 for all n N. In ths example the left-hand sde n (55) s equal to one for every n, but by geometrc ergodcty the rght-hand sde n (55) s less than one. 6 Example [GE SGP] Consder the MC wth state space and transton kernel p(0, 0) = and Ω = {0} {(a, b) : a, b {,,..., a}} p((a, b), (a, b )) =, for b, p((a, ), 0) =, p(0, (a, b)) = The nvarant dstrbuton π can be calculated to be (a+) : a = b 0 : otherwse. π(0) = and π((a, b)) = (a+) for b {,,..., a}. (56) Häggström [3] has shown that ths MC s GE wth δ 0 =. In order to prove that k n = 0 for all n N, t suffces to show that k = 0 (see [5]). Ths can be seen as follows: Defne A n,n = {(n, n), (n, n ),..., (n, )} and A n, = {(n, )}. Then we have k k(a n,n ) = n (n+) n (n+) A n,n p(, A c n,n )π() π() = n (n+) n. (57) A n, Lettng n yelds k = 0. 06

17 Kontoyanns and Meyn [4] have proved that geometrc ergodcty and SGP are not equvalent usng the same example, but a dfferent argument based on an Lyapunov functon approach. Häggström [3] orgnally used the example n order to present a sequence of random varables connected to a geometrcally ergodc MC wth fnte second moments but not followng the central lmt theorem. In fact, ths result mples that the MC cannot satsfy SGP, snce by a theorem due to Cogburn [] for every sequence of random varable connected to a Markov chan satsfyng SGP and havng fnte second moments the central lmt theorem holds. We now show that We start from the observaton δ 0 = δ = δ =. p n (0, 0) = n N. (58) Defne d(0, (a, b)) = a b + (a, b) : a, b {,,..., a} and d((a, b), 0) = b (a, b) : a, b {,,..., a}. Usng equalty (58) t s not dffcult to see that for all n d(0, (a, b)) we have But ths mples that for n d((a, b), 0) = b p n (0, (a, b)) = π((a, b)). (59) p n ((a, b), ) π T V p n b (0, ) π T V π({(a, b) : d(0, (a, b)) > n b, a, b {,,..., a}}) C b n for some C > 0. (60) Ths yelds that s an upper bound for δ 0. To see that s also a lower bound, note that p n (0, ) π T V p n (0, (n +, )) π((n +, )) = π((n +, )) = 4 n. Next we show that δ. Smlar calculatons as n (59) yeld for all ε (0, ] and n d((a, b), 0) = b p n ((a, b), ) π T V C ( ε) b + ε n for some C > 0. Snce f defned by f ((a, b)) = ( ε) b s n L (π), the desred nequalty follows. To see that s also a lower bound for δ, we calculate k n (A n ) for A n = {0} {(a, a) : a {,,..., n}}, n. It s not dffcult to show that p (0, (j, j)) = π((j, j)) j N. 07

18 Ths mples Applyng (58) and (6) we obtan k n (A n ) = = = = Now t can be easly deduced that p k (0, (j, j)) = π((j, j)) j N, k. (6) π(a c n ) π(a n )π(a c n ) π(a c n ) π(a c n ) π(a n ) π(a c n ) π(a c n ) π(a n ) A n p n (, A n )π() n =0 n p n (0, A n )π() π(a n )π() =0 +π(n )p(0, A n ) + π(n) π(a c n ) π(a c n )π(a n) [π(a n) π(a n )(π(n ) + π(n)) +π(n )p(0, A n ) + π(n)] = π(a n)(π(n ) + π(n)) + π(n )p(0, A n ) + π(n) π(a c n )π(a n) = + p(0, A n) 3π(A n ) π(a c n )π(a π(n) (6) n) lm k n(a n ) n =. Apply nequalty (5) of Theorem 8 to conclude that δ. Altogether we have now shown that δ 0 = δ = δ =. Note that the nfma δ and δ are not attaned but the nfmum δ 0 s. Acknowledgement. Ths research s part of a project that s supported by the Deutsche Forschungsgemenschaft. References [] Cogburn, R.: The central lmt theorem for Markov processes. In: Proc. Sxth Berkeley Symp. Math. Statst. Probab., : 485-5, 97. [] Döbln, W.: Sur les proprétés asymptotques des mouvements régs par certans types de chaînes smples. Bull. Math. Soc. Roum. Sc., 39(): 57-5; (): 3-6, 937. [3] HÄGGSTRÖM, O.: On the central lmt theorem for geometrcally ergodc Markov chans. Probab. Th. Relat. Felds, 3: 74-8, 005. [4] KONTOYIANNIS, I., MEYN, S.P.: Geometrc ergodcty and the spectral gap of non-reversble Markov chans. ARXIV: ,

19 [5] LAWLER, G.F., SOKAL, A.D.: Bounds on the L spectrum for Markov chans and Markov processes: a generalzaton of Cheeger s nequalty. Trans. Amer. Math. Soc., 309: , 988. MR [6] LEVIN, D.A., PERES, Y., WILMER, E.L.: Markov Chans and Mxng Tmes. Amercan Mathematcal Socety, 008. MR [7] MEYN, S.P., TWEEDIE, R.L.: Markov Chans and Stochastc Stablty. Sprnger, 993. MR87609 [8] NUMMELIN, E.: General Irreducble Markov Chans and Non-Negatve Operators. Cambrdge Unv. Press, 984. MR [9] NUMMELIN, E., TUOMINEN, P.: Geometrc ergodcty of Harrs recurrent chans wth applcatons to renewal theory. Stoch. Proc. Appl., : 87-0, 98. MR [0] REED, M., SIMON, B.: Methods of Modern Mathematcal Physcs. Volume I: Functonal Analyss. Academc Press, New York, 97. [] ROBERTS, G.O., TWEEDIE, R.L.: Geometrc L and L convergence are equvalent for reversble Markov chans. J. Appl. Probab. 38(A): 37-4, 00. MR9553 [] ROBERTS, G.O., ROSENTHAL, J.S.: Geometrc ergodcty and hybrd Markov chans. Electron. Comm. Probab., : 3-5, 997. MR4483 [3] SALOFF-COSTE, L.: Lectures on Fnte Markov chans. Lecture Notes n Math. 665, Sprnger, Berln, 996. MR [4] WÜBKER, A.: Asymptotc optmalty of sopermetrc constants. To be publshed n Theoretcal Journal of Probablty, DOI: 0.007/s [5] WÜBKER, A.: L -spectral gaps for tme dscrete reversble Markov chans. Preprnt, 0. [6] WÜBKER, A.: Spectral theory for weakly reversble Markov chans. Preprnt, 0. 09