5 Birkhoff s Ergodic Theorem

Transcription

1 5 Birkhoff s Ergodic Theorem Amog the most useful of the various geeralizatios of KolmogorovâĂŹs strog law of large umbers are the ergodic theorems of Birkhoff ad Kigma, which exted the validity of the strog law to statioary sequeces. These are of importace eve i the cotext of radom walks (that is, sums or products of idepedet, idetically distributed radom variables), because may iterestig quatities associated with radom walks ca be oly expressed as fuctioals of the radom walk paths that caot themselves be decomposed as sums of ide- pedet radom variable. 5. Measure-preservig trasformatios ad statioary sequeces Defiitio 5.. Let (,F,P) be a probability space ad let T :! be a measurable trasformatio. The trasformatio T is said to be measure-preservig if for every A 2 F, or equivaletly if for every radom variable X 2 L, P(T (A)) = P(A), (5.) E(X ± T ) = EX. (5.2) The triple (, P, T ) is the said to be a measure-preservig system. A ivertible measurepreservig trasformatio is a ivertible mappig T :! such that both T ad T are measure-preservig trasformatios. Exercise 5.2. Let A be a algebra such that F = æ(a ). Show that a measurable trasformatio T :! is measure-preservig if equatio (5.) holds for every A 2 A. Example 5.3. Let = {z 2 C z =} be the uit circle i the complex plae, ad for ay real umber µ defie R µ :! by R µ (z) = e iµ z. Thus, R µ rotates through a agle µ. Let be the ormalized arclegth measure o. The each R µ is measurepreservig. µ a b Example 5.4. Let T 2 = R 2 /Z 2 be the 2 dimesioal torus. For ay 2 2 matrix A = c d with iteger etries a,b,c,d ad determiat, let B A : T 2! T 2 be the mappig of the torus iduced by the liear mappig of R 2 with matrix µ A. The B A preserves the uiform 2 distributio o T 2. NOTE: i the special case A =, the mappig B A is sometimes called Arold s cat map, for reasos that I wo t try to explai. Example 5.5. The Shift: Let = R (the space of all ifiite sequeces of real umbers), ad let T :! be the right shift, that is, T (x 0, x, x 2,...)= (x, x 2, x 3,...). 53

2 It is easily checked that T is measurable with respect to the Borel æ algebra B, defied to be the smallest æ algebra that cotais all evets {x : x 2 B}, where B is a oedimesioal Borel set. (Here x = (x 0, x,...), ad x is the th coordiate.) If is a Borel probability measure o R, the the product measure o B is the uique probability measure such that my (B 0 B B m R R ) = (B i ) for all oe-dimesioal Borel set B 0,B,... (The existece ad uiqueess of such a measure follows from the Caratheodory extesio theorem.) It is easily checked (exercise) that the shift T preserves the product measure. The otio of a measure-preservig trasformatio is closely related to that of a statioary sequece of radom variables. A sequece of radom variables X 0, X, X 2,... is said to be statioary if for every iteger m 0 the joit distributio of the radom vector (X 0, X,...,X m ) is the same as that of (X, X 2,...,X m+ ) (ad therefore, by iductio the same as that of (X k, X k+,...,x k+m ), for every k =,2,...). Similarly, a doubly-ifiite sequece of radom variables (X ) 2Z is said to be statioary if for every m 0 ad k 2 Z the radom vector (X k, X k+,...,x k+m ) has the same joit distributio as does (X 0, X,...,X m ). Statioary sequeces arise aturally as models i times series aalysis. Useful examples are easily built usig auxiliary sequeces of idepedet, idetically distributed radom variables: for istace, if Y,Y 2,... are i.i.d. radom variables with fiite first momet E Y i <, the for ay sequece (a ) 0 satisfyig P a <the sequece is statioary. X := X a k Y +k k=0 Clearly, if T is a measure-preservig trasformatio of a probability space (,F,P), ad if Y is a radom variable defied o this probability space, the the sequece X = Y ± T (5.3) is statioary. This has a (partial) coverse: for every statioary sequece X 0, X,... there is a measure-preservig system (,P,T ) ad a radom variable Y defied o such that the sequece (Y ± T ) 0 has the same joit distributio as (X ) 0. The measurepreservig system ca be built o the space (R,B ), usig the shift mappig T : R! R defied above. This is doe as follows. Suppose that (Y 0,Y,Y 2,...) is a statioary sequece of radom variables defied o a arbitrary probability space (,F,µ). Let Y :! R be the mappig Y(!) = (Y 0 (!),Y (!),Y 2 (!),...). 54

3 This is measurable with respect to the Borel æ algebra B (exercise: why?), ad the iduced probability measure P = µ ± Y (that is, the joit distributio of the etire sequece Y uder µ) is ivariat by the shift mappig T (that is, T is P measure-preservig). By costructio, the joit distributio of the sequece Y = (Y 0,Y,...) uder µ is the same as that of the coordiate sequece X = (X 0, X,...) uder P. This is a useful observatio, because it allows us to deduce theorems for the origial sequece Y from correspodig theorems for the sequece X: i particular, lim! i= X Y i = E µ Y 0 a.s.-µ () lim! X X i = E P X 0 a.s.-p. (5.4) Observe that if T is measure-preservig the for every itegrable radom variable Y, i= EY = E(Y ± T ). (5.5) This ca be proved by followig a familiar lie of argumet: (a) if Y = B is a idicator variable, the equatio (5.5) reduces to the defiitio of a measure-preservig trasformatio; (b) if (5.5) is true for idicators, the it is true for fiite liear combiatios of idicators; ad hece (c) for oegative fuctios (5.5) the follows by the mootoe covergece theorem. It follows from (5.5) that compositio by a measure-preservig trasformatio T preserves all L p orms; i fact, for p = 2 eve more is true. Propositio 5.6. Let T :! be a measure-preservig trasformatio of a probability space (,F,P). The the mappig U : L 2 (P)! L 2 (P) defied by Uf = f ± T is a liear isometry. Proof. This is a trivial cosequece of equatio (5.5). 5.2 Birkhoff s Ergodic Theorem Defiitio 5.7. If T is a measure-preservig trasformatio of (,F,P), the a evet A 2 F is said to be ivariat if T A = A. The collectio I of all ivariat evets is the ivariat æ algebra. If the ivariat æ algebra I cotais oly evets of probability 0 or the the measure-preservig trasformatio T is said to be ergodic. Defiitio 5.8. A measure-preservig trasformatio T of a probability space (, F, P) is said to be mixig if for ay two bouded radom variables f, g :! R, lim Ef(g ± T ) = (Ef)(Eg).! Exercise 5.9. Show that if T is mixig the T is ergodic. 55

4 Exercise 5.0. Show that if A is a algebra such that F = æ(a ) the T is mixig if for all A,B 2 A, lim! E A( B ± T ) = P(A)P(B). Exercise 5.. Let T be the shift o (R,B, ) (See otes for defiitios. The probability measure is the product measure; uder the coordiate variables are i.i.d. with distributio.) Show that T is mixig, ad therefore ergodic. Theorem 5.2. (Birkhoff s Ergodic Theorem) If T is a ergodic, measure-preservig trasformatio of (,F,P) the for every radom variable X 2 L, lim! X ± T j = EX. (5.6) The proof will follow the same geeral strategy as the proof of a umber of other almost everywhere covergece theorems, icludig Kolmogorov s SLLN ad Lebesgue s Differetiatio Theorem. The first step is to idetify a dese subspace of L for which the covergece ca be easily established. Defiitio 5.3. Let T be a measure-preservig trasformatio of (, F, P). The a radom variable Y 2 L 2 (,F,P) is called a cocycle (more properly, a L 2 cocycle) if there exists W 2 L 2 such that Y = W W ± T. Propositio 5.4. If T is a ergodic, measure-preservig trasformatio of (, F, P) the the cocycles are dese i the subspace of L 2 cosistig of all X 2 L 2 such that EX = 0. That is, for ay such X ad ay " > 0 there is a cocycle Y = W W ± T such that kx Y k 2 < ". Proof. Let L 2 0 the closed liear subspace of L2 cosistig of all X 2 L 2 with mea EX = 0, ad let V be the liear subspace of L 2 0 cosistig of all cocycles. By elemetary results i Hilbert space theory (see, for example, W. RUDIN, Real ad Complex Aalysis, ch. 4), it suffices to prove that if X 2 L 2 0 is orthogoal to V the X = 0 a.s. Cosider first the special case where T is ivertible, with measure-preservig iverse T. Let V Ω L 2 be the set of all cocycles ad let V be its L 2 closure. We claim that if X is orthogoal to the set V the X = 0 a.s. To see this, observe that for ay cocycle W W ±T, EX(W W ± T ) = EXW EX(W ± T ) = EXW E(X ± T )W = EW(X X ± T ). Sice this holds for every W 2 L 2, it holds for all idicators, ad cosequetly the radom variable X X ±T = 0 almost surely. This shows that X is a ivariat radom variable. 56

5 Sice the measure-preservig trasformatio T is ergodic, it follows that X is (almost surely) costat; sice EX = 0 it follows that X = 0 a.s. The geeral case requires a additioal bit of Hilbert space theory, specifically, the fact that ay Hilbert space is self-dual. This implies that ay liear isometry U : L 2 0! L 2 0 has a uique adjoit mappig U : L 2 0! L2 0, i.e., a mappig such that for ay two elemets X,Y 2 L 2 0, hux,y i=hx,u Y i, (5.7) where hx,y i=exy is the ier product. The existece of U Y follows because X 7! hux, Y i is a bouded liear fuctioal, ad so self-duality (the Riesz-Fisher theorem) implies that there is a uique elemet U Y such that equatio (5.7) holds for all X 2 L 2 0. It is the easily checked that U is liear. Here we will specialize to the liear isometry U iduced by T, that is, UX = X ± T ; see Propositio 5.6 above. (Observe that whe T is ivertible, the adjoit of U is just U X = X ± T.) Suppose, the, that X 2 L 2 0 is orthogoal to the space V of all cocycles; we must show that X = 0 a.s. If X is orthogoal to V, the for every Y 2 L 2 we have 0 =hx,y Y ± T i =hx,y UYi =hx,y i hx,uyi =hx,y i hu X,Y i =hx U X,Y i, ad cosequetly X U X = 0, sice it is orthogoal to everythig (ad i particular, to Y = X U X, which implies hx U X, X U X ). Now cosider X UX: sice X = U X, ad so X = UX almost surely. To do so, we will show that kx UXk 2 =hx UX, X UXi =kx k 2 +kuxk 2 2hX,UXi = 2kX k 2 2hX,UXi = 2kX k 2 2hU X, X i = 2kX k 2 2hX, X i = 0, kx X ± T k 2 =kx UXk 2 = 0. This will the imply that X is almost surely equal to a T ivariat radom variable X, ad sice the oly ivariat radom variables are costat, it will follow that X = 0 a.s. 57

6 It is trivial to check that the ergodic theorem (5.6) holds for L 2 cocycles. I particular, if Y = W W ± T the Y ± T i = W ± T i W ± T i+, ad so Y ± T i = W W ± T! 0. Sice by Propositio 5.4 the space of cocycles is dese i L 2, at least whe T is a ivertible measure-preservig trasformatio, it is also dese i L, ad so there is a dese subspace of L for which (5.6) holds. Propositio 5.5. (Wieer s Maximal Ergodic Lemma) Let T be a measure-preservig trasformatio of (,F,P). The for ay radom variable Y 2 L ad ay Æ > 0, P ( sup Y ± T i Æ ) E Y Æ. (5.8) Proof. Without loss of geerality, Y 0 ad EY > 0. For each iteger m defie F m to be the evet ( )! : max Y ± T j (!) Æ. m Fix k, ad defie B = Bm k (!) ( bad ) to be the set of all itegers r km such that T r (!) 2 F, that is, such that oe of the first m ergodic averages startig at time r is at least Æ. Claim: The set Bm k is cotaied i the uio of a collectio of o-overlappig itervals J µ [km+ m] such that X Y ± T j (!) J Æ. (5.9) j 2J Proof. Proceed left to right i the iterval [km+m] util reachig the first iteger r 2 B k m. By defiitio of the set B k m, there is a iterval J of legth J m with left edpoit r such that iequality (5.9) holds for J = J. Now proceed iductively: assumig that J l is defied, let r l+ be the smallest iteger i B k m to the right of J l, ad let J l+ be the smallest iterval of legth J l+ m with left edpoit r l+ such that (5.9) holds. Cotiue i this maer util reachig the right edpoit km of the iterval [0, km]. Give the Claim, the rest of the argumet follows routiely. I detail, the oegativity of Y implies that Æ Bm k mk+k X Y ± T j. 58

7 Takig expectatios o both sides, we deduce that for ay m, P ( max m Y ± T j Æ ) EY Æ. Fially, let m!ad use the mootoe covergece theorem. Proof of Birkhoff s Theorem. By Propositio 5.4, every X 2 L 2 with mea EX = 0 ca be arbitrarily well-approximated i L 2 orm by cocycles, that is, for ay " > 0 there is a cocycle Y = W W ± T such that kx Y k 2 < " 2. By the momet iequality (i.e., Hölder), By Wieer s Maximal Iequality, kx Y k kx Y k 2 < " 2.! P supø (X ± T i Y ± T i ) Ø > ") ". But as we have already see, the ergodic theorem holds for ay cocycle, so the ergodic averages for Y coverge to 0 almost surely. Hece, P lim sup Ø X ± T i Ø Ø Ø ")! ". Sice " > 0 is arbitrary, this proves that relatio (5.6) holds for every X 2 L 2 with mea EX = 0. It follows trivially that (5.6) holds for every X 2 L 2. It remais to show that (5.6) holds ot oly for radom variables X 2 L 2 but also radom variables X 2 L. This ca be doe by trucatio, with aother use of Wieer s Maximal Iequality. (Exercise!) 59