‘Jl.irl-.vtu$..\ 7. {{3} (4:14! 1. (€31 Y..l\. . 1, :31...) (-21... Kitegirz x. J I . 5213...: LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE MSU is An Affirmative Action/Equal Opportunity Institution c:\circ\datedm.nrn}p.1 CONVERGENCE OF STOCHASTIC PROCESSES INDEXED BY PARAMETERS By Jongsig Bae A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Statistics and Probability 1993 Shlomo Levental, Adviser ABSTRACT CONVERGENCE OF STOCHASTIC PROCESSES INDEXED BY PARAMETERS By Jongsig Bae The convergence of a stochastic processes indexed by parameters which are ele- ments of a metric space is investigated in the context of an invariance principle of the uniform central limit theorem (UCLT) for a sequence of martingale difference random variables. We assume the integrability condition on the metric entropy with bracketing which implies the total boundedness of the underlying metric space. We study the con- vergence of a partial-sum process which depends on both time and parameters. A eventual uniform equicontinuity result for the process is developed which essentially gives the invariance principle of the UCLT for a sequence of martingale difference random variables. This result generalizes the UCLT for a sequence of i.i.d. random variables of Os- siander (1987). We directly prove the invariance principle while Ossiander (1987) stated it by quoting the result of Dudley and Philipp (1983). This result also gen- eralizes that of Levental (1989). Levental (1989) developed a method to get uniform central limit theorems for martingale differences that are uniformly bounded. We remove the uniform boundedness requirement by using chaining argument with strat- ification adapted from that of Ossiander (1987), which, in turn, adapted from that of Bass(1985) in the context of set-indexed partial-sum processes. An application to Markov Chains is given. This generalizes those of Gordin and Lifsic (1978) and Levental (1989). ACKNOWLEDGEMENTS I would like to express my deepest appreciation to my advisor Professor Shlomo Levental for his guidance throughout the preparation of this dissertation and his concern for my work. His vision of Probability always guided and inspired me. I would like to thank Professor Roy Erickson, Professor Joshep Gardiner, Professor Hira L.Koul, and Professor Habib Salehi for serving on my guidance committee. I also would like to thank faculty members and secretaries in my department for helping me in many ways. Finally I would like to share my joy of finishing this degree with my family, my friends, and some faculty members in the department of mathematics. To my wife Kyunghee, to my daughter Jieun, and to the memory of my late sister Yunseon iv Contents 1 Introduction 1 1.1 Invariance Principle of CLT for Martingale Differences - One dimen- sional case ................................. 2 1.2 Proof of Theorem 1 ............................ 3 2 Invariance Principle of UCLT for Stationary Martingale Differences 15 2.1 Introduction and main result ....................... 17 2.2 Proof of Theorem 2 ............................ 28 3 An Application to Markov Chains 51 3.1 Introduction ................................ 51 3.2 Invariance Principle of UCLT for Markov chains ............ 55 Chapter 1 Introduction The convergence of a stochastic processes indexed by parameters which are elements of a metric space is investigated in the context of an invariance principle of the uniform central limit theorem (UCLT) for a sequence of martingale difference random variables. In Chapter 1 we consider an invariance principle of the central limit theorem for an array of martingale difference random variables. This is one dimensional case of the Theorem 2 in Chapter 2. In other words an element of the metric space is fixed in the context of the invariance principle of the uniform central limit theorem for martingale differences. In Section 1.1 we formulate an invariance principle of the central limit theorem for an array of martingale difference random variables. In Section 1.2 we prove Theorem 1 1 using two different methods. The proof of Theorem 1 is organized as follows. Firstly, the weak convergence of the finite dimensional distributions is discussed. Secondly, the eventual uniform equicontinuity is proved using two different methods. The Method I includes an inequality appears in Brown (1971). After using a stopping time the inequality reduces the eventual uniform equicontinuity to the convergence of the finite dimensional distributions. For the Method II we construct a sequence of stopping times with which the condition in a maximal inequality ( Lemma 3) is fulfilled. Then the eventual uniform equicontinuity of the stopped process is reduced to the convergence of the finite dimensional distributions. 1.1 Invariance Principle of CLT for Martingale Differences - One dimensional case Let {firm : n 2 1,i 2 0} be an array of 0‘- fields on a probability space (0, T, P) such that for each n andi Z 1, fnJ—l _C_ fnflu We assume that {5 fm : 1 g i g n,n Z 1} n,i’ is an array of martingale difference random variables. Define conditional variances V1131: En,i—1( 2 ):: E( 2 lfnj—l) n,i n,i for i = 1, ...,n. Define, for every t 6 [0,1], [rzt] X11“) 2: 2 {11,23 i=1 int] V,,(t) := Z‘w (1.1) i=1 where [x] is the integer part of x. For each to, Xn() is an element of the space D[0, 1] of all functions on [0,1] which have left limits and are continuous from the right. Let B denote the standard Brownian motion on [0, 1]. If the distribution on Xn(-) converges weakly in D[O, 1] to GB for some nonnegative a, then {Em} is said to have the invariance principle of central limit theorem (CLT). The following Theorem 1 states an invariance principle of CLT for an array of martingale difference random variables. Throughout this dissertation indicator functions are denoted by sets whenever it can cause no ambiguity. Theorem 1 Assume that (a) for everyt 6 [0,1], Vn(t) —> t in probability, and (b) for every 6 > 0, the sum: En,,-_1({,2,,,-[|{n,,-| Z 6]) 2:1 converges in probability to zero (a Lindeberg condition). Then Xn(-) converges weakly in D[0,1] to B. 1.2 Proof of Theorem 1 Finite dimensional convergence of Xn A single time point case is a consequence of Theorem 1 of Pollard (1984, p 171). For a sequence of real-valued random variables {Yn : n 2 0} we denote K, i Y0 to mean the weak convergence of K, to Y0. Let 0 = to 3 t1 3 S tk = 1 and let a = (a1, ...,ak) E Rk. Suppose, for each i = 1, ...,k, that Xn(tg) => B(ti). Then by the usual Cramer Wold argument, the weak. convergence of the finite dimen- sional distributions of {Xn(t) : t E [0,1]} to those of {B(t) : t 6 [0,1]} will follow from k k Zaan(t,-) => Zen-8(a). (1.2) i=1 i=1 For the proof of (1.2) we note that [nil] [M2] 1: Zaanfti) = (01+"'+01k)an,i+(a2+°~+01k) Z {W- i=1 i=[nt1]+l [ntk] +“’+ak 2: {mi izfntk—li'f’l [ntl] [Rig] [ntk] = an,.,1+ 2 wn,.-,2+---+ 2 10nd,], i=1 i=[nt1]+1 i=[ntk_1]+l where rum-,1 = (a; + + 005m“ for [nt1_1] < i S [ntz], l = 1,...,k. The sum of conditional variance of this martingale difference array converges in probability to (01+"°+ ak)2t1+(a2 + ' ‘ ' + ak)2(t2 — t1) + ' ° ° + Giftk — tk—l) which is the variance of 2le aiB(t;). For the Lindeberg condition of {mm-,1} we note 4 that, for every 6 > 0, the sum [nt1] ' ° ° ' 2 En i- 2 ' ni 6 (01+ +ak) {=21 , 1( n,z[i€.l> lal+"'+akl]) 2 If] 2 n 6 1 + (a + . o o + ak) Enai- ( n: {nail > ) 2 i=[nt1]+l 1 ’ '02 + - - . + Gk] Wk] 2 6 + ' ' ' + a: Z En,¢'—1( n,i[i€n.i| > I |]) i=[ntk_1]+1 0”“ converges in probability to zero by the Lindeberg condition of {67”.}. Applying the single time point result of Theorem 1 to the array {tum-,1} we have the weak convergence of the finite dimensional distributions of {Xn(t) : t 6 [0,1]} to those of {B(t) : t 6 [0,1]}. Eventual uniform equicontinuity of Xn We will prove the eventual uniform equicontinuity of X7, using two different meth- ods. Method I Define k Tn = max{k : Z um,- S 2}. 2:1 Interpret Tn as zero if ”12.1 > 2. Because V,”- is fn,,_1—Iiieasurable, the event {i S Tn} is fn,;_1-measurable. This implies that Tn is a stopping time. Note that En,i—1({n,il‘rn Z i]) = [Tn Z ilEn,i—1(§n,i) = 0. 5 This means that a family of random variables {am-[7,, 2 i]} is a martingale differ- ence array. Let us define [nil Zn(t) 2: Xn(t /\ Tn) = Zgnnan Z 2], 1:1 and ~ [nil Vn(t) := Vn(t /\ Tn) = ZE,,,,-_1(€,2,,,-[Tn Z i]). i=1 Note that P{Z,,(t) 7b Xn(t) for some t 6 [0,1]} [M] [at] = P{Z {n,,-[‘r.,, _>_ i] 75 :5,“- for some t 6 [0,1]} 3 P{[nt] > Tn for some t 6 [0,1]} S P{n > Tn} = P{Z Vn,i > 2}. i=1 By the condition (a) the last probability converges to zero. This implies that the process Zn(-) have the same asymptotic behavior as the original process Xn(-). Note that l6n,i[Tn 2 i]] S lénfllo Then for every 6 > 0, we have [nt] 2 En,i—1(€7Zz,if7-n _>. iifléndan Z 2]] Z 6]) i=1 6 Int] E Z En,£—1(€721,iil€n,il Z 6])- i=1 By the condition (b) the last sum converges in probability to zero . Note also that P{Vn(1) 75 Vn(1)} S P{n > Tn} = P{i um- > 2} i=1 which converges to zero by (a). This implies that we have the weak convergence of the finite dimensional distributions of {Zn(t) : t E [0, 1]} to those of {B(t) : t E [0, 1]}. Next we will prove the eventual uniform equicontinuity of Zn. We need to show that for every 6 > 0 there exists 6 6 (0,1) such that lim sup P{ sup IZn(t) — Zn(y)| 2 26} S c. (1.3) lt—ylS5 By looking at Billingsly (1968, 'p 56), we observe that (1.3) follows from : For every 6 > 0 there exists 6 E (0,1) such that limsup P{ sup IZn(s) — Zn(t)| Z c} S 66. (1.4) n tSsSt+6 The proof of the next Lemma 1 appears in Brown (1971). Lemma 1 Let U0 2 0, U1, U2, - - ' ,Un be martingale. Then, for every c > 0, (c’llUnl — 2) s / c“|Un| P Ui >2 < P U71 > {11%| | c}_ {I I OH {MN} {|U,.|>c} Now note that, by Lemma 1, Pf sup lZn(8)—Zn(i)| Z 26} tSsSt-HS [n8] 2 P{ sup I: (“[73, Z 1]] > 26} t<3l]l>c} I: [nt]+1 |/\ Observe that [n(t+6)] { Z {anTnle} 1=[nt]+1 is uniformly integrable since 23;, um,- < 2, and [n(t+5)l Z €71,1[7' 21]? N(O,(S) l=[nt]+1 by the finite dimensional convergence. Therefore we have l1msupP{ sup lZn (t )— Zn(y)l 2 26} It— yl<6 1 -<- “l 6 {IN(0,5)|>€} six/3 / . lN(0,6)l e {|N(0,1)I>7;} 1 /6 _fi = — 56—] ac 2du 2r {lul>7‘g} u e 2 2du c2\/27r /{|u|>7‘-}2 < 66. IN(0,5)I |/\ The last inequality is true for 6 small enough since we can choose 6 so that 1 c2\/27r {|u|>7‘3} 2 “2 ue 2du < e. This proves (1.4). That completes the proof of Theorem 1 by using Method I. Method II We begin with the following Lemma 2. Lemma 2 If V,,(t) ——> t in probability for everyt 6 [0,1], then there exist a sequence of stopping times {an} and a sequence of constants {c,,} such that sup |Vn(t) — t] S 6,, a.s.. OStSan Proof of Lemma 2 Claim 1 For each 6 > 0 there exist a sequence of stopping times {on} such that P{on = 1} —+ 1, and l sup ]V,,(t) — t] S c a.s. for — < E. OStSan n 2 For the proof of Claim 1 we define k k k 0,, = max{— : | E va' — -— Interpret on as zero if the set is empty. Note that ”mi is fn,;_1-measurable, the event {on = f} is fn,i_1-measurable. This shows that on is a stopping time. Note also that P{an < 1} = P{an < 3—} = P{IZun,,~—1|>-2€—}. i=1 The last term converge to zero since Vn(1) —+ 1 in probability. This means that P{on = 1} ——> 1. We will Show that for ~71; < % sup |V,,(t) — t] S e a.s.. OStSan We assume without loss of generality that g < t < knil for £7111 S on. By the monotonicity of Vn, we have = v.( >— —— n n n n k-i-l k+l l S lvn( )— |+_ n n < 6+6 2 2. Similarly we have t — Vn(t) S 6. Therefore for 0 S t S on we have |Vn(t) - tl S e. 10 In other words we have for :11— < 3- sup |Vn(t) — t] S c a.s. OStSan We have finished the proof of Claim 1. Now we choose integers n(l) < n(2) < - -- such that l l P{adfi) < 1} < 5,; for n 2 n(k). Set 6,, = 21—]: and an 2 0,45%) when n(k) S n < n(k +1). Then we h ave 1 SUP Iva“) — t] S SUP anU) — t] S '— OStSan 0950(2)?) 21: This proves Lemma 2. Now we choose {an} and {6"} as in Lemma 2. Note that the event {an _>_ i} is fn,,-_1-measural)le. Then {gm-[an _>_ fl} is an array of martingale difference random variables. Let us define mt) ;= Xn(t /\ an), and <> n(t) :2 Vn(t /\ an). Note that P{Yn = X,,} 2 P{ozn =1}——)1. 11 This implies that, as before, the process Yn(-) have the same asymptotic behavior as the original process Xn(). Now note that n i = - > — . génnlan _ n] Since i léndfan Z 5]] S lgndla we have the Lindeberg condition for {gm-[an 2 fl}. Since P{an : 1} —> 1, we have P{Vn(an) = Vn(1)} -—> 1. Therefore we have Vn(1) = Vn(an) —> 1 in probability. This shows the weak convergence of the finite dimensional distributions of {Yn(t) : t 6 [0,1]} to those of {B(t) : t 6 [0,1]}. The proof of the following Lemma 3 appears in Pollard (1984, p 177). Lemma 3 (Pollard. 1984, p 177) Let {Z(t),.7:t : 0 S t S b} be an L2- martingale with conditional variance process V. If , for every t, then P{ sup |Z(t) — 2(0)) > 6) s 3P{|Z(b) — 2(0)) > g). ogtgb C2 Now let 6 > 0 and let 6 S — For n large enough so that 26,, S s 6 [i6, (i + 1)6], we have 24' E(Vn((i+1)6) — V..(s)|r.) S + + _<_ S E(Vn((i+1)6)—((z'+1)6)/\ anlf.) E(((i+1)6)/\ an — s /\ anlf.) 13(3 /\ a. — V..(s)|r-.) 2cn+6 where .7, is defined by ft 2 ft): for tk S t < tk+‘l- Lemma 3 apply. By working with Yn(t) instead of Zn(t) in Method I , we have limsupr sup lYn(t) - Yn(y)| Z 66} S |/\ l/\ It-yIS5 If] lim sup: P{ sup |Y,,(s) — Y,,(i6)| Z 26} 1%] 2 i=0 n .=0 :69:va limsup3P{Yn((i + 1)6) — Y,,(i5)l Z c} 1%] 322P{IN(0,6)I > e) i=0 3(1 + [%1)P{IN(0.6)I > .} 13 l l 1 _fi — 3(1 + [3]) Q—W/{|u|>7‘3}e du 66 62v27r {Iu|>;§3} <6. 2-93 ue 2du |/\ The last inequality is true for 6 small enough since we can choose 6 so that 66 62 \/27r {Iul> 7‘3} 2_fi ue 2du < 6. This proves the eventual uniform equicontinuity of Yn. That completes the proof of Theorem 1 by using Method II. 14- Chapter 2 Invariance Principle of UCLT for Stationary Martingale Differences In Chapter 2 we investigate an invariance of the uniform central limit theorem for a sequence of stationary martingale difference random variables. Our setting in Chapter 2 involves ergodicity and stationarity. We do not use ergodicity in order to estab- lish Theorem 2. We use the stationarity in one place of the proof of Theorem 2. However, by inspecting the usage of the stationarity we observe that the restriction can be removed using the Lindeberg condition which is essential condition in a finite dimensional central limit theorem for a sequence of martingale differences. See, for example, Brown (1971) for a discussion of the Lindeberg condition. By these reasons Theorem 2 can be considered as a proof of a sequence of martingale differences. We 15 remark that Greenwood and Ossiander (1991) discuss this problem in a more abstract setting. In Section 2.1 we formulate an invariance principle of the uniform central limit theorem for a sequence of stationary martingale difference random variables. The formulation includes metric entropy with bracketing. We assume the integrability of the metric entropy with bracketing which implies the total boundedness of the un- derlying metric space. We define a partial—sum process which depends on both time and parameters. In Section 2.2 we prove Theorem 2. We truncate the random vari— ables at level of fn and construct the partial-sum process of the truncated random variables. The content of Proposition 1 is essentially the eventual uniform equicon- tinuity of the truncated partial-sum process. We prove Theorem 2 using Proposition 1. The organization of the proof of Proposition 1 is as follows. By using condition (b) in Theorem 2 we can construct a stopping time which does not restrict us. This allow us to apply Freedman inequality (This is Bernstein inequality for martingales). Then we use the chaining argument with stratification for the truncated and stopped partial—sum process. The chaining argument with stratification basically means that we use the usual chaining argument in each strata. This is originated by Bass (1985) in the context of the set indexed partial-sum process. Then Ossiander (1987) adapted it in the context of i.i.d random variables. We adapt the steps in Ossiander (1987) to our setting of a martingale difference random variables. We obtain the invariance 16 principle which is uniformity in time using Lemma 3. 2.1 Introduction and main result Let B be a o-field on a set S. We choose (9 = SZ, 7 = 32, P) as our basic probability space. We denote by T the left shift on Q. We assume that P is invariant under T, i.e., PT‘1 = P, and that T is ergodic. We denote by X = ---,X_1,X0,X1,--- the coordinate maps on (I. From our assumptions it follows that {X,},-€Z is a stationary and ergodic process. Next we define for each i E Z a o-fields M,- := 0(X, :j R:f€M,-andf€L2(Q)}. We also denote for each f E L2(D) Ei—1(f) == E(f|M.-_1), and H0@H_1 :2 {f6H0:E(f-g)=0foreacthH-1}. Finally for every f, g E L2(Q) we put d(f,g) == [130‘ - 9)2ll/2. '17 We assume .77 Q H0 9 H_1. From our setup it follows that for every f E f, { f (Ti(X )), M;} is a stationary martingale difference sequence. It follows from Theo- rem 1 that for every f E .7: Int} —\/1= 2 f( (T'(X )converges weakly in D[0,1] to B n .lZl[Jf"] i: 1 where B is the standard Brownian motion on [0,1]. For every f E .7: and for every t E [0, 1], we define [1zt] Sn( %2f( V.) (2.1) where and [x] is the integer part of :13. Our goal of this Chapter 2 is to find sufficient conditions for the existence of an invariance principle of uniform central limit theorem (UCLT). This essentially means showing that £(Sn(f,t) : f E fit E [0,1]) —-> £(G(f,t) : f E .77,t E [0,1]), where the processes that are involved here are indexed by .7: (8) [0,1] and are considered as random elements in B(f' (X) [0, 1]), the space of the bounded real—valued functions on f8) [0,1], taken with the sup norm. (C(f, t) : f E f,t E [0,1]) is a Gaussian process 18 which is continuous in both (f, t) a.s. and ,for a fixed f, G(f, ) is a Brownian motion and, for a fixed t, G(-, t) is a Brownian bridge. Next we define the metric entropy with bracketing. See, for example, Dudley (1984). Definition 1 For (f, d) we define the covering number with bracketing 118(6, .7, cl), or VB(6) if there is no risk of ambiguity, as the smallest n for which there exists {fé’5,f&5,-~,ff,’6, 3,5} g H0 so that for every f E .7 there exist some 0 S i S n satisfying (1) i145 _<_ f S fitfb and (2) d( 1!,67 :5) < 6‘ Define the metric entropy with bracketing to be 113(6) :2 HB(6,f',d) := 1n VB(6,f,d). We also define the associated integral for 0 < 6 S l 6 I J(6) := / [HB(u)]§du. 0 We use the following notations: 19 For a function (,0 : .7: —+ R, we let llrllr == 8111) MM (22) fEf denote the sup of |<,o| over .7. We write I] - H in stead of I] - II}- when there is no risk of ambiguity. We also let llrlls == 83]) MD - 99(g)| (2-3) denote the sup of |c,o(f) — 99(g)| over (6) where (5) == {(f,9)€f> 0 such that P*{ sup in: Ei—llf(vi) " 904)] 2 2 D —> 0. f.g€Ho i=1 "d2ffagl } Then for every 6 > 0 there is 6 > 0 such that limsup P*{ sup ||S,,(-,t)||5 Z 6} S 6. (2.4) n 03th 20 In the following Corollary 1 we state an invariance principle of UCLT. We write 5 := .7: (8) [0,1]. Define a pseudometric p on S by 10((f19t1)? (f2,t2)) = max{d(fltf2)? ltl — t2|}- It is well known that 8(8) is complete in the sup—norm, so that (3(5), II - “8) form a Banach space. We use the following definition of weak convergence due to Hoffmann- Jorgensen (see Hoffmann-Jorgensen, 1984, p 149). Definition 2 A sequence of B(S)-valued random functions {Yn : n _>_ 1} converges in law to a B(S)-valued random function Y, denoted Yn => Y, if E900 = 1im E*g(Yn),V9 E 613(5)) ll ' Ils), 1).—>00 where C(B(S), HHS) is the set of all bounded, continuous functions from (3(5), HHS) into R. Here E’“ denotes upper expection. Corollary 1 Under the assumptions of Theorem 2, 5,. => G. G(f,t) is a Gaussian process with EG(f,t) : 0 and EG(f1,t1)G(f2,t2) = (t1 /\ t2)Ef1(X)f2(X) which is uniformly continuous in both (f,t) a.s. with respect to ,0. Proof of Corollary 1 Finite dimensional convergence of 5,, 21 Let 0 = to S t1 S S tk =1 and let or = (01-1, ...,ak) E 12’“. We need to show that (Sn(f17 t1), ' ' ° 1 Sn(fk1tk)) => (C(fla t1), ' ' ' , fok, tk))° Then by the Cramer Wold argument this will follow from 201871 (f17ti) iZ(’Y3G(.f17ti) For the proof this we note that k [nil] Zaisn(fiati) = 732%?le “'+01kfk(Vi)} —[‘nt2] .I. g 2 {02f2(I/i)+”'+akfk(vi)} n i:[’nt1]+1 l [ntk] + ---+— 2 crime) fii=[n1k_,]+1 : 24.71.13 say) i=1 where {CM} is an array of martingale difference random variables. The conditional variance of 21;] oz,S,,(f,-,t,-) converge in probability to Efzaifjw +(t2 ‘_ t1» Midi/AV +(tk _ tk-1)E(al2cflf(V))1 which is the variance of 2le oz,- G(f,-,t ) For the Lindeberg condition of {C11 i} we first note that I01f1+°"+ 0111-ka S ([01] +°' ' + [Gk-Ill?- Then we note that for every 6 > 0 ZEi—l( 3,1llCn,1| > 6I) 22 |/\ + [ntl] h; Ei—lffalflfvi) + ‘ ' ' + akfk(V,-))2[Ia1f1(Vi) + ' ' ' + akka/‘ll > fie” [ntg] h 2 Ei—1{(0l2f2(Vi) + - ° - + akfk(vi))2fla2f2(vi) + ' ' ' + akfk(Vi)l > WEI} i=[nt1]+l [ntk] ...+ l 5; E.-.{aZfz(V.-)Hakfk(v.-)I > V661} 7" i=1nt.-.1+1 (1a.) + . - . + lakl)2 W” 2 W6 Ei— F Vi F Vi > a +”_+ a 2 [M2] 726 (l 2' ' .1) z; E._1{F2(W)[IF(V.-)I > f I} n fiwlm I012] + - - - + IOIkI a2 [nth] n6 ...+ _k X E,_1{F2(V.)[|F(V.-)l > l/—|]}' i=[ntk_1]+l ak Then by the Lindeberg condition of F(V,-), which is given by stationarity, the sum 1: ZEi-1(Cwi,ill 61) i=1 converges in probability to zero . This is the Lindeberg condition for {Cm}. Apply- ing the single time point result of Theorem 1 to the array {Cw} we have the weak convergence of the finite dimensional distributions of {Sn(f,t) : f E .7, t E [0,1]} to those of {G(f,t) : f E .7, t E [0,1]}. Eventual uniform equicontinuity of 5,, We need to show that the following : For every 6 > 0 there is 6 > 0 such that limsupm sup IS..(f,s)—Sn(g,t)IZe}Se. " d(f.g)S5ils-tls5 23 Let m 2 1 and let 6 = %. By Theorem 2, we can choose 62 and n2 such that for all n 2 n2 1 1 P7 Sn at _ Sn at Z _ S —' {0833?1 «[851; 62 | (f ) (g )| 3m} 3m Since .7 is totally bounded, .7 is covered by finite numbers of rig—balls. That is, there exists an fk = fk,m E .7 such that d(f,fk) < 62 and d(g,fk) < 62. By Theorem 1, there exist 61 and n1 such that for all n _>_ n1 1 3m. P*{ sup IS.(1.,s)—S.(f).,t)12 $13 Is-tIS51 Let 6 = min{61, 62}. Note that SUP [Sil(f15)_5n(gvt)l d(f)9)_<_5vls-‘IS5 S 811p sup lSn(f13)'—Sn(fk13)l US$51d(f.fk)S5 + $111) ISn(fkaS) — Sn(fkat)i 13-1156 + SUP SUP |3n(fk,t)-Sn(g,t)l- 03‘51d(fk.g)35 Then, for all n _>_ max{n1, n2}, we have P*{ sup Is. a} = 0. Indeed, note that V6 > 0, 36 > 0 such that 1imsupP*{ sup lSn(f1,t1)—Sn(f2,t2)|26}Se. 7‘ p((f1,t1)v(f2.t2))S5 Let A be the finite set of the 6-nets. Then, by the finite dimensional convergence, we have lim lim sup P*{sup ISn(fo,,ta)I > a} = 0. a-‘(XD n 0644 We write Mn := supaeA ISn(fa,ta)I. Then note that ”£7an E Mn + sup I5n(f1,t1) —- 5n(f2,t2)| a.s.. p((f1.t1),(f2,t2))ga Then we have 011130 lim sup P*{IIS,,II5 > a + 6} 25 S limsup P*{IIS,,II5 - Mn > 6} + (111120 lim sup P*{Mn > a} <6. Letting 6 —> 0, we get the eventual boundedness of {Sn}. Remark 2 : {Sn} is eventually tight. I.e. V6 > 0, 3 a compact set K such that lim sup,, P*{Sn E G} < 6 for all open sets G so that G 2 K. Indeed, the eventual uniform p-equicontinuity and the eventual boundedness to- gether imply the eventual tightness of {Sn} (see Andersen and Dobric (1987), Theo- rem 2.12). Remark 3 : Apply Theorem 7.11 (case 3, Remark (1)) in Hoffmann-Jorgensen (1984) to conclude that Indeed, we consider ‘1! := {(32). “3* :a;c E 12,51. E 5,and Zaksk is a finite sum} 1: where eizk “*s“(t) 2: eizk (111(31). Then \II is a selfadjoint semigroup of bounded, continuous complex-valued functions on 8(5). By the finite dimensional convergence of {Sn}, we have that 11,111] EMS”) exists W) E \II. 26 If t1 75 t2, then we can find if) E \I' such that ib(t1) 75 w(t2). Also we can find 3 E 5 such that t1(s) 71$ t2(s). Choose a 76 0 so that —7r < at1(s) < it and —7r < at2(s) < 71'. Then eia3(t1) :: eiatif-S) 7Q eiatzf-S) : 6ia8(t2). This shows that W separates points in 8(5). Remark 4 : See Theorem 4.1 in Andersen and Dobric (1987) for the uniform continuity of G. We observe that the assumption (a) in the theorem implies the total boundedness of the metric space (.7, d) and the assumption (b) is an asymptotic Lipschitz condition in the average sense with a Lipschitz constant D. Theorem 2 can be considered as a generalization of Theorem 3.1 of Ossiander (1987). To specialize our work to their framework we assume that P 2: (P0)Z for some P0, a probability measure on S (in other words the X,- are i.i.d) and that all the functions in .7 depend on X1 coordinate only and E(f(X1)) = 0. In that case the condition (b) in Theorem 2 boils down to the Lipschitz condition (2.3) in that paper. We note that we prove the invariance principle directly by using Lemma 3 while they state that by quoting the result of Dudley and Philipp (1983). Theorem 2 can be also considered as a generalization of Theorem 2 of Levental (1989) in the following sense. We remove the uniform boundedness requirement of underlying martingale difference sequence. We note that the condition (b)(i) in that 27 paper is weaker than our condition (b). The other two conditions (a) and (b)(ii) together are similar to our condition (a) about the integrability of metric entropy with bracketing. We also note that we use stationarity in one place in the proof of our Theorem 2 while it was not used in Levental (1989). 2.2 Proof of Theorem 2 For a > 0, let a ifa0, nZ 1,tE [0,1] ande.7, let WW) = 2121mm» and [nil SSW) = if; Proposition 1 Assume that (b) there exists a constant D > 0 such that 28 ;{f‘“’“’(%) — E._1(IWQ>(V,-))}. (2.6) P. { sup : Ei—llf( (V)_ g( Vill2 2 D} _)0. f,gEHo ._1 "d2(f19) Then for every 77 > 0 ,for every 6 > 0, and for each 0 <- TWP/2 10101331115531111911171161)+na))saim—1221mm) 12.7) k=0 where K is a universal constant and Lx = ln(x V e). Proof of Theorem 2 Fix 1} > 0. The family {f() : f E .7} is uniformly bounded by an envelope F(') == $131,111) 6 112(9) because VB( 1) < 00 and )< Z] lfl,(1)()|+ If."1()l) 6 L202)- Note that {f(V )} IS a martingale difference sequence. So we have lEi—l(f(Vi)1{|f(V.~)|>\/770})l = IEi—l(f(vi)1{|f(V.~)Ig\/E6})l' Note also that for 0 > 0, f E .7, and t E [0,1], we have 8111) 811p lSn(f, 1) - S'f.9)(f,t)l 05151 fef 11111 a = SUP SUP— WIZUM fW (V) 0f1} +sup— $521514 1( V')l{lf(V1)l<\/_9})1 feff +sup\/_ZlE1 1(\/7—19)111f1v1)1>\/‘6})l S %ZF(V1)11F1141>1/1711 +sup— 22E1—( 1 (W l-)1{lf(V) |>\/_9}) g $331414 )1{F(V)>\/—9} QnZE1_(1(Vi)(l/1{F)>fi0}) S 9—,; 1: F( )111F V1)>¢‘6} +—ZE1— 1F2 (V1)1 {F(1V)>\/Re}~ The last two terms converges in L2(Q) to zero because the stationarity and Dominated Convergence Theorem together imply _ZEW Vz)1{F(V)>\/7—16}) = E(F2(V)1{F(V)>\/z—19}) = 0(1). 30 Therefore we have P{ sup |l3n(-,t) — stew-J)“ > i} = 0(1). 05th Since 811p ||5n(',t)||a = sup ||5£9)(°,t)lls+2 sup l|5n(-,t)-5£9’(xt)ll, ogc<1 0951 03th it remains to Show 6 6 P{ SUP ||3£9)(wt)||6 > -} S -. 03th 2 2 We may choose 77 so that 13 Zexp{—n2Lk} S 5. k=0 2 Now choose 6 small enough so that 1(¢E(J(6) + 776) g 5. Then by Proposition 1 (2.8) is true for 0 S amr/z and 11 large enough. End of proof of Theorem 2. Proof of Proposition 1 We define a stopping time Tn, for n 2 1 k - . _ . 2 m z: n/\max{k_>_0: sup ZE1—1[f(‘/‘) 9(Vz)l LgéHo i=1 716120, 9) < D}. Then "from (b) we get P{Tn < n} -—> 0 as n -—> 00. We write 0,1(t) :2 [(nt) /\ Tn], the integer part of (nt) /\ Tn. 31 Note that an(t) S an(l) for each t 6 [0,1]. Note also that “‘1’ E._1[f(V.-) — g(V.-)]" P sup _>_ D =0. 2.9 {figEHo g nd2(f,g) } ( ) We write 1 an(t) S$S’(f.t> = —— Z {f‘m’U/i) -— E.-1(f<fi9>(v.-))}. (2.10) J73 i=1 Since P{Tn < n} —> 0 as n —> 00, it is enough to prove that for every 77 > 0 , for every 6 > 0, and for each 0 fl amwz P*{ sup ”Sm-.0“. 2 K\/5(J(6) + 775)} :13 Zexm—wflk} (2.11) 0931 k=0 In order to prove the last inequality we follow the steps in Ossiander (1987). Step 1 : Fixn>0,andfix6>0. Fork20,let and k B 7!: = Z H (51')- 1:0 Let {ak : k 2 0} be a strictly decreasing sequence with limk_.oo ak = 0. The values of ak will be specified after applying Freedman inequality below. We write 1k I=[(lk+1,ak) and I; I: [ak+1, 00). 32 Note that the intervals Ik is a partition of the interval (0, a0). Step 2 : Fix 0 S 929. We construct a nested sequence of upper and lower 6k—approximations to flfio) in 142(5)) in the following way . For f E f, let (2.12) 9 where ij is the i that satisfies (1) and (2) in the Definition 1 for 63-. Let unk() (ZIW uk( (D and — Mx/EfiJi-(J). (2.13) Note that In k and un k depend on .77 only through to 50, - -- ,ka 6k and f0 60, -- 3; 5k respectively. Observe that sup If feyr MEN.” < aofi _ 2 9 (rm/77 SUP lun,k( )l V |1nk( )|_ IET and SUP lun.k(-) - l,-..;.( )l < am/T—l (2-14) f6? 33 Note that for each k 2 0, ln,k(‘) S f(fi9)() S un,k(')7 (2.15) and 0 S un,k+1(') _ ln,k+l(') S un,k(') _ ln,k(')' (2'16) Step 3 : We construct the sets with which we partition 3:3?- Choose kn so that nakn+1 < (J(6) + ”(SM/5 g nakn. (2.17) For 0 _<_ k S kn, define the following subsets of the sample space () _ ln.k(') An,k(f) = [UM W e 11.], (2.18) and ” : [un.k(') _ ln.k(') A...k(f) fl 6 11]. (2.19) The sets {Bn‘k(f) : 0 S k S kn + 1} are partitions of the sample space induced by the sets {Anyk(f),0 S k 3 kn} 2 to :3 , b 11 3s : Er : / 31>. : b = Ami-(m A.,.(f).13k3k.. (2.20) and B.,..+. = (Q 8.141)) For k 2 1, let kn-H = 11 am. (221) 1:1: Since Cn,k(f) C flfl,k_1(f), we have, on the set Cn,k(f), ln,k(') — ln,k—1(') S Un,k—1(°) *ln,k—1(°) S akfi- (2.22) Step 4 : In this Step we stratify 3131”,” using the partition {Bn,k(f) : 0 S k S kn + 1} constructed in Step 4. Recall that 0,.(21) 2 [(nt) /\ Tn(t)], the integer part of (nt) /\ Tn(t). ForOSkSkn+1,let ”at“ S..,k(f,t)=—;{;{f1‘/_WV)IBM(1)(V)— —1(f(‘/_9V)(V)13...(f)(V))} and (1) 1 071(1) LTn,k(f7t) = W: {1n,( k( V)1B,. k( (n(V) — Ei—i(1n,k(Vi)1Bn,k(f)(Vi))}- Then, since 6 S 92-11, kn+1 313(11): ZS..(M, (2.23) 35 ForOSkSkn, IS‘rn,k(f7t)_ L(1),k(f t)| 1 W“ 3 W ZEUS/.0“ V)—ln ,k( W)}18n,k(f)(Vi) 1 0,.(t) ”)0 + 77-; §E1_1({f(‘/_)V( Vi)—lnk( i)}1Bn,k(f)(Vi)) 1 an(1) S 77;" 1:1 {uwz,k(l/i)_ln,k(l/l)}1An,k(f)(l/1) 1 0,;(1) + W i=1 Ei—1({un,k(Vi) - ln,k(Vz‘)}1An,k(I)(Vi)) Likewise, we have ISTn, kn+1(f’t )_ LTn 1)k‘n'i'1(’f’ t)| 1 an(1) S W {:1 {un,kn+1(Vi) — ln,kn+1(l/i)}an,kn+1(f)(l/i) 1 an(1) + 7n.— :1 Ei—1({un,kn+l(l/i) - ln,kn+1(Wllan,kn+1(1)(Vi)) = Rig/cum”) + RigianUl (225) Note that, on the set B,,,kn+1(f), we have un,kn+l(‘/i) — ln,kn+1(Vi) \ffi akn+1 («1(5) WNW/17 Tl |/\ l/\ 36 So we have 8‘13, mm 3 («1(6) + now, (2.26) and Rfiiikmn 5 (1(6) + now. (2.27) Step 5 : Now, on the individual Bn,k(f)’s, we compare each lower (fig-approximation, Ink, to the lower (So-approximation, 171,0. For each f E .7: and t E [0,1], let 0n(t) L) k=0 kn+1 1 an“) = I; W {:1 EWWW)-ln,j—1(V))1B,,,,(I)(V)) — Ez—1((ln,J(V.) an—:(W))lsnkm(V)))} kn-H on“) Step 6 : We now compare Sig) to LS2) defined above. Combining (2.24),(2.25) and (2.29), we have for each f E f, t 6 [0,1], and for 0 S 329, 153Z>—L33’w(f, m kn+l s I Z {Swarm — L31’,k(f,t)}l k-O kn“ 1 0 + I;{L3,.,’.(f —L,,3’(ft>}l kn+10 0 S ZRinikU) k=0 kn+1 + ZRSIM) k=0 kn+1 + Z: IR3i{k(f,t)I. k=1 38 Therefore we have sup ”9‘3 )(-t t)lla OStS S sup HLW t)lla OStS + 2 sup ||S33’(-,t) — L33’(-,t)lla OStSl SUP lle ,.)(t t)lles OStS + 2ZIIR3§1 kn l/\ + 2ZIIR3‘ikH k:+1 + 2 z: sup H1233 (, )II. (2.30) k: l 0 (K + 4)(J(5) + n5)\/5} O_t_l < W 08111: ||L33)(nt)lla > 2770} + ;P{HR(T(,: ),k||>77k0)} + 2":P{IIR3:’,kn > 723"} -—o kn+1 + Z P{ Os 272(2)} (2.32) k: _ 39 The values of the constants 170, Nico), 775.1), and 77,?) will be specified later. Step 7 : The individual terms of equation (2.32) above are bounded using Freedman in- equality and the upper bound of the cardinality of Ufzo 55(6j) where f(6) :2 {f(l,6a fab ' ' ' 7fr£3(6),63 f:B(6),6}‘ Fix f E .77. Take Vii?) = Uzi—:13. Then we have P{R3E’lkw) 2. 123”} = P{ak+1R(¢(,:),k(f)2 06:} an(1) : P{ak+17—; Z; Ei—1({un,k(vz') _ ln,k(Vi)}1An,k(f)(Vi)) 2 05,3} an(1) 1 P{; 2 Ei—llun,k(‘/i) _ ln,k(‘/i)l2 2 06:} i—l l/\ 0,.(1) 1 H; g: E._1[f§;(V.-) — fé.(V.-)]2 2 062} |/\ a.(1)Ei_1[fi;;‘(V.-)- f§.(Vi)12 s H; nd2(f§k,f§‘k) 2 D} on“) E._1[f(Vi) — 9W2 S P{filégo i=1 ndzUfl) 2 D} : 0 (2.33) where we used (2.9) in the last equality. 40 Since 333),]. depends on .7 only through the (at most) exp{2'yk} members of ULO f(6J-), we have k" (0) (0) k" (0) (0) Z P{IIRT.,k|| > m. ls Z exp{27k}||P{RT.,k(-) > m. }H = 0- k=i) k:d) Step 8 : The proof of the following Lemma 4 appears in Freedman (1975). Lemma 4 (Freedman Inequality) Let (di)ISiSn be a martingale diflerence with respect to an increasing o-fields (fi)0SiSna i.e. E(d.-|.7:}_1) = 0,i 2 1,~-,n. Suppose that lldil|00 S M for a constant M < oo,i = 1,-~-,n. Let T S n be a stopping time relative to the (.75) that satisfies lIZZ=1E(d?|f;-_1)Hoo S L for a constant L. Then for each 6 > 0 2 Pilédil > 6} S 2€XP{—m}- For 0 S k S kn, RS)“ f) is a sum of nonnegative random variables each bounded by oh, and note that R£:)k(f) S 77)?) a.s.. Note that R31’,.(f> — R33’,,.(f) 1 0,.(1) : W :1{{umkfl/i)—ln,k(Vi)}1An,k(f)(Vi) _Ei—1({un,k(vi) _ ln,k(Vi)}1An,k(f)(Vi))} 0,.(1) 2: E ahsay, i=1 where Ei_1(oz.-) = 0. Note also that lail S 2a;c a.s.. 41 Using the algebraic inequality (a — b)2 S 2(a2 + b2) and noting the calculation of (2.33) we see that on(1) 2 BMW?) S i=1 Take 77,?) fEJ'", = 211)?) 2 «n(l) ‘7; Z E._1[(un,k(%> 471.4%)? i=1 3 206,3 :2 L a.s.. . Note that D6}: = ak+1 77:0) S akniol. Then by Lemma 4, for each P{R31’,.(f) > 123"} S |/\ S S P{R3iikm — R32{.(f) > n3" — n3°’} 0,;(1) Pi Z 0i > Uzi-0)} i=1 2 "30) } (20513 + Zak???) O 2 m‘.’ 2(2ak’7ic0) + 2akn3‘”) 0 IL 80k ' exp - { 2 exp{- exp{— Hence, since R321: depends on .7: only through the (at most) exp{27k} members of Usz ‘7:( 6.7 ) 3 kn. 1 1 ZP{IIR3J,.II > 713’} k=0 kn s ZeXp{27k}IIP{R§le-(')>17i1)}|l k=0 42 77(0) 2 eXp{27k — —} S < iex {2 — 06,3 _ k=0 P 7k 8a Isak-H k 2 " 06k S Zexpi27k— 7:} k=0 S ZeXP{-n2L/€} k=O where D :6. V2. . . (lk k(8(27k+n2L/€)) (2 34) Note that the strictly decreasing sequence {ah} in (2.34) is chosen so that Step 9 : Note that for 1 S k S kn + 1, f E f, Rm k(f, 1) is a sum of martingale difference sequence. So we may write 0,;(1) 1232.0) == R3130, 1) := Z a say, i=1 where E;_1(fi,-) = 0. By (2.22), we have |flzl S 2a,c a.s., and by the similar argument as in Step 8, we get on(1) Z Ei—lWi) 2 an(1) E Ei— 1[(u u,—nk 1( (Vi)—ln,k-l(‘/i)l2 i=1 43 3 206,11 :2 L a.s.. Take 17(2): .Note that 77(2)— — 77):)“ and of. < 6L1 Again by Lemma 4, for each f E .7, we have P{R3iikf (f )> 2732’} 0,;(1) : P{ ; fli > 771:2) (2)2 77k _<_ 2exp{- 2(206,3_ +2akn£2)) (2)2 "k = 2exp{— } 2(2akn3 +2akn33’) m9) = 2 _— exp{ 8a).} 062. = 2exp{— 8,34} k D862 S 26XP{-—} Note that Rfiilk also depends on .7: only through the (at most) exp{27k} members of Uk_0 H( 3). So we have km (2) (2) Z P{HR ,kll > ”k } k—l kn'l‘l 062 s 2 : exp{Qn- 7:} —l S 2 Z exp{—-7)2Lk}. k=O Step 10 z 44- Now we will use Lemma 3 to come up with the sup over t in the first and the forth terms of (2.32). We write on(t) Vn(t) :2 Z Ei_1(fl,-2). Then note that v..(1)— m) s V1.0) s 206,3-.. (2) 2 Claim 2 2135,:1 g (Elk—l. 12 . D52 Since 1],?) = ——a’;’—‘ and ak = (SAWHYH, we have 20(5),:1 _ 3 < 1 (217(2))2 — 16(27;c + 7ka) i 12 So Lemma 3 apply. Therefore we have H0832 IRiilkU, t)| > 27253)} S 3P{|R§:),k(f)l > 7793- (235) Therefore we have 2 2 P{ sup ”33.1.0.3” > 2723 ’} 03:51 kn+1 D62 S 6 Z exp{2wc — ——2k k=1 80”: S 6 Z exp{—772Lk}. (2.36) k=0 Finally, for f,g E f, note that L3?.’(f, t) — L3?)(g, t) on(t) z V11? 313.4%) — 13.4%) — E._1—Ii,.(m>} 45 where [£0 and 13m L353.’(f) — where Ei_1(C,-) : 0. Note that lCil S 2a0 a.s.. are [mo corresponding to f and g respectively. We write, as before, on(1) Lflww=aWLn—LW =§3asw, i=1 When d(f,g) < 6, we have 10"“) _ 722 E11 [1f10( V- )_lgo(v,))2 10,,(1) S - "EH/3&4 1( --.(/(Vi))2}1/2 +wmmnm—tflmwfl +{E1-l(ui]io(v)_ l;".o(V))}1/2l2 10,.(1) S _ ”ZliEi—( 1 -—-g(V,-))2}1/2 Haaa(V)—AJW»P“ +{E.-_1(93‘0(W) - 930(Vi))2}1/2l2 30”“) S _ :2 Ei- llf(V —gV( ¢)l2 30:0) +;;Emmwo 770} 0,.(1) ‘ = P{ZCi>7Io} i=1 7702 } (18062 + 2a0770) - S Zexp{-2 Therefore we have P{IIL‘TflM-ms > no} 2 S 2 “M47" " 20806727: 200770)} 2 S. 2 “M47" _ 2(2aonc7: 200770)} = 2exp{4‘ro — 8—7230} 2 2exp{4’ro " 98—22:} = 26(pr _ 90628882):$2 + 722)} 47 = 2 eXP{470 — 9(270 + 722)} = 26XPi“'14’70 ‘— 9772} l/\ 2 end-772} |/\ 2 Z exp{—7]2Lk}. k=0 We write, as before, an“) Vn.0(t) 3: Z Ei—1(C.'2)- i=1 Then note that Vn,0(1) _ l/n,0(t) S l/n(1) S 18062 Claim 3 18D62 S %. Since 710 = 9—3—3: and a0 = (HWY/'2, we have 18052 _ 1 < 1 (2717322 — 2470 So Lemma 3 apply. Therefore we have P{ SUP IIngltatHls > 2770} S 3P{|L(T?,)(f) - nglwll > 770}- (2-37) 0931 Therefore we have P{ SUP ||L§3)(-,t)lla > 2770} S 6 Zexp{-772Lk}- (2.38) 03th k=0 By (2.32), it remains to show that (2.31) holds for some fixed constant K. Note that k" (0) Z 77k k:0 48 We write where f: = X: : kn 2 D 6 k=0 ak-H D 5: 568 2M + was +1))1/2 (SH-101” 8D1/2 :3 6k—1(7k + 77214,“)1/2 5k 3201/2 X 6,427,, + 77%)”? k=1 3201/2 Z 6427;” + nLl/zk) k=1 32(2D)1/2 Z 61,7,1/2 k=1 3201/2n26kL1/2k k=1 [ml/216 := 66, k=0 . By definition of 7],, 26ml” 3 :1):;( 6)]1/‘226. = 226W )1?” s 4j§1/6j:I[HB(u)]‘/2du = 4 A6[HB(u)]1/2du = 4J(6). 49 Therefore we have 2 mi”) 3 4 - 32(20)‘/2J(6) + 3201/27766 = (128\/§ + 32&)(J(6) + "ax/E, Recall that 1 0 O nl.)=77l-), andnll=nl31 Then we have 9062 Go 9D52x/§(2'm + 772)"2 .56 = 9\/§\/l36(270 + 772)”? 770: g 36x/13(671/2 + 776) = 36\/-D_(6[fIB(U)]1/2 + 116) g 36(J(6) + n6)\/5. Therefore we have kn kn+1 260+2Zn£°’+2§_:0n,fi”+4 Z 17,?) k: 0 = 2770+2Z77ic0)+4:::77icm +4 2 film 1:: O < K(J(6) + 7,6)x/5 where K = 72 + 128\/2+ 3205. We have shown that (2.31) holds. This completes the proof of Proposition 1. 50 Chapter 3 An Application to Markov Chains In Chapter 3 we give an invariance principle of the UCLT for a stationary Markov chain as an application of Theorem 2. Theorem 3 generalizes the results of Gordin and Lifsic (1978) and Levental (1989). 3.1 Introduction Let X0, X1, X2, - -- be a strictly stationary and ergodic Markov chain taking values in a measure space (S, B) with transition mechanism P(:1:, dy) and initial distribution a. We assume there exist 1 S b so that P($,dy) = P(:v,y)a(dy),xay E S (3.1) 51 and 0Sp(:c,y) Sb 0 as n ——> oo 53 where P Ar) 8 — P A P B W): sup |( ) ()()| AEo(Xo),B€o(Xn),P(A)>O P(A) According to Ibragimov and Linnik (1971, p.367-368) we will have this by verifying a sufficient condition (Doeblin’s condition) for sci-mixing : Let :1: E S and let 6 = > 0. _1_ (2+1 If a(A) S 6, then P666) = I. Fwy) = I. pa s 66(6) :1- c. This observation together with ergodicity imply the cp-mixing condition of the {Xi}. Moreover , 99(n) satisfies 90(n) S 2Cr", where C > 0 and 0 < r <1. Let p(n) :2 sup E(P”f(Xn))g(X0). ||f||=1rl|9||=1JEMrgEM Then we have llP"|| = SUP lanfll = 10(6), llfll=1rfEM where the operator P is restricted to M. We recall (Ibragimov and Linnik (1971), Theorem 17.2.3, p.309 ) that M") S 2 99(7?) This shows that ||P"'|| —> 0 exponentially fast. This will lead to limsupn(||P"||)1/" < 1. Therefore we have the representation (I — P)‘1 = :0 P on M . This completes the proof of Lemma 5. 54 From the stationarity, using the Kolmogorov consistency theorem, we may assume that the process (X,) is double sided . Also each f : S —> R will be considered as f : Q = S2 —-> R by putting f((X,-)) 2: f(X0), so the invariance principle of UCLT in the following theorem has the same meaning as in Chapter 2. 3.2 Invariance Principle of UCLT for Markov chains The following Theorem 3 is a generalization of Theorem 3 of Levental (1989). We remove the uniform boundedness requirement in the Theorem 3 of Levental (1989). We also include invariance principle of UCLT as in Theorem 2. In the proof of Theorem 3 we use the same idea as in Levental (1989), Lemma 2 which is adapted from Theorem 1 of Gordin and Lifsic (1978). Theorem 3 Let .7 be a class of real valued functions defined on S. Assume that {3.1) and (3.2) hold. If 1 / [1n 6301f, L2(o))]1/2do < oo, 0 then the Invariance Principle of UCLT for .7: holds. Proof of Theorem 3 Let g = {(I—P)‘1[f—Eaf] : f E .7}. Observe that f—Eaf E M. We assume without loss of generality that Eaf = 0, f E 7". Then Q can be rewritten as {(1 — P)’1f : f E .7}. For every g E g we define g : S2 -> R by 55 §(X) = g(X0) — Pg(X_1). Observe that Pg(X_1) = E{g(X0)|X_1}. Then {6 : g E g} Q H0 @ H-1. This means that {§(Ti(X))} is a martingale difference for each g E g. Recall that d(f,g) = [E(f — g)2]1/2, f, g E L2(Q). We need to show that: For every 6 > 0 there is 6 > 0 such that [nt]-— 1 limn"‘supP { sup sup WI}: (f1 —— X)| > e} S e. (3.5) 0 0 there is 6 > 0 such [nil-1 limn"supP {sup sup 0 0, P{ SUP C(Xpufl > V56} 0931 = P{ma \/7—z€} = P{Q G(X.-) > We} 2 nP{G(X1) > We} 2 0(1) by the Dominated Convergence Theorem. Our goal is to prove (3.6) by using Theorem 2. Condition (a) : Let u > 0. Suppose ||f1 — f2|| S 11. Since (I — P)g = f, then we have ”91 — 92“ S “(I — P)‘1|| - u. So we have d(j1,j2) S -||([ — P)'1|| - 11. Next we need to construct the brackets for {£1}. Write VB(u) 2 I/B(u,}', L2(a)). By definition of VB(u), there exists {fé’wfafim - - - ’frl/B(u),u’ f:B(u),u} so that for every f E fthere exist someO S i S V801) 57 satisfying (1)./1,11. S f _<_ :1” and (2) d(fil’u, :11.) < u. We write VB(u) :2 VB(u, {fl},d) :2 VB(u,f,L2(a)). Let g E {[7}. Correspond g E g to g. Note that g = (I — P)'1f for some f E .7. We construct {96,u?gg,uv ' ° ' 3913(u),u3g;‘3(u),u} g H0 in the following way : Fix 0 S i S VB(u) satisfying (1) z'l,u S f _<_ fzifin (”211(2) d(fil,u9fii,‘u) < u‘ First we note that (I—P)“f = fl”! 11:0 N 00 = 2 m + 2 m 11:0 n=N+1 N 00 S E Pnfu + Z Pnf, N will be specified later. n=0 n=N+1 Next we note that from (3.4), with M := supfef Hf”, Stlpl Z P"f($)| = SUPIPZP"f($)| :L‘GS n=N+1 IES n=N g 12“ Z P"f|| from (3.4) n=N = bIIPNU- P)"‘f|| S bIIPNII'||(1-P)"||°||f|| 58 S b'M'||(1—1D)—1||'||1DN|| S constant-HPNH = CN (say)- Since ||PN|| ——> 0 as N —> 00, we can choose N large enough so that 2cN < u. Define, for j = 0,---,VB(u), N l l 9m := Z Pnfjm — CN n=0 and N gxu I: Z Puffin +CN. 11:0 So for g = (I — P)‘1f and film S f S fffu we get: fithSgh and d(gi,u3g:u) ngu _ gin” N S 26N + H 2 P"(fi‘u -' in)” n=0 N _<_ 2CN + || Z 10"“ ' ”(L-1ft — film)” 7120 S C - u, where C 2 1+ supN “SQ/=0 P"||. Similarly we construct the brackets {Pgé’w P93”, - - - Pg£3(u),ngf/B(u),u} for {Pg : g = (I—P)‘1f, f E f}. Recall that §(X) = g(X0)—Pg(X_1). We now define the brackets 59 {§é,u,§a‘,u, ~ ~ - ,§L8(u),u,§;‘3(u,,u} g Ho for g by the following equations éiu == 9§,u(Xo) - PgMX—l), and £3. == gZu(Xo) - Pg§,u(X-1)- Then from the inequalities gi,u(X0) S 9(X0) S gifu(X0) and Pgiu(X— 1) < P9(X—1 ) < PQZJX—l) we have 95,..(Xo) - PgEfu(X—1) S 9(Xo) - P9(X-1) S 93f..(Xo) - ng,u(X—1) and d2(§f-,m§.‘fu) = ”(923. - 91.)” + ||P(93fu - gf,..)|| = 2||(gi‘,.. - 9.2.)“ S 2011. We conclude that utuna>==/Wm#mturnnw s/WWw——WfL(m%u = 2C/2?[lnz/B(u,f,L2(a))]%du 0 <00. Condition (b) : First assume that ”P” < 1. We choose D 2 CW +1. Note that m/gm wxy mwyD} 91,916Ho.=1 n (91,92) “ bd2(g"1 9‘2) < P* su ’ ~ ~ 2 D - Ema; 1— ||P||2)nd2(91,92) } 6 ___—ED l—IIPIP } : p{ = 0. Applying Theorem 2 , we have (3.8). If HP” 2 1 then there exist N so that IIPNII < 1. We will work with N Markov chain (Xk+N1)?;O, k = 0,1,~-,N — 1. Note that [nt]—1 N_1[nt]— 1 [v-1 [nt]—l IZKX' )I=IZ ZN Xk+NillSZIZflXk+Nill i=0 k=0 i: 0 k=0 i=0 61 Let 6 > 0. Choose 6k , for each [C = 0, - - - N — 1, so that [nil-1 limSUPP {SUP SUP Vii: (f1— f2)( )(Xk+Ni)l Z 6} S 6} S 6- 0c} 5} < N5} 05 <6 2% n {O