An Indicator Function Limit Theorem in Dynamical Systems

Transcription

1 arxiv: v2 [math.ds] 5 Nov 2009 An Indicator Function Limit Theorem in Dynamical Systems Olivier Durieu & Dalibor Volný October 27, 208 Abstract We show by a constructive proof that in all aperiodic dynamical system, for all sequences (a n ) n N R + such that a n ր and an n 0 as n, there exists a set A A having the property that the sequence of the distributions of ( a n S n (l A µ(a))) n N is dense in the space of all probability measures on R. This extends the result of [5] to the non-ergodic case. Keywords: Dynamical system; Ergodicity; Sums of random variables; Limit theorem. AMS classification: 28D05; 60F05; 60G0. Introduction and result Let (Ω,A,µ) be a probability space where Ω is a Lebesgue space and let T be an invertible measure preserving transformation from Ω to Ω. We say that (Ω,A,µ,T) is a dynamical system. Further, the dynamical system is aperiodic if µ{x Ω : n,t n x = x} = 0. It is ergodic if for any A A, T A = A implies µ(a) = 0 or. ForarandomvariableX fromωtor,wedenotebys n (X)thepartialsums n i=0 X Ti, n. The present paper concerns the question of the limit behavior of partial sums in general aperiodic dynamical systems. In 987, Burton and Denker [2] proved that in any aperiodic dynamical system, there exists a function in L 2 0 which verifies the central limit theorem. In general, for functions in L p spaces, Volný [7] proved that for any sequence a n, LaboratoiredeMathématiquesetPhysiqueThéorique,UMR6083CNRS,UniversitéFrançoisRabelais, Tours; olivier.durieu@lmpt.univ-tours.fr Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, Université de Rouen; dalibor.volny@univ-rouen.fr

2 a nn 0, there exists a dense G δ part G of L p 0 such that for any f G the sequence of distributions of S nk (f) is dense in the set of all probability measures on R, see also Liardet and Volný [6]. This work is also related to the question of the rate of convergence in the ergodic theorem (see del Junco and Rosenblatt [3]). In Durieu and Volný [5], a similar result is shown for the class of centered indicator functions l A µ(a), A A and for ergodic dynamical systems. The following theorem was obtained. Theorem Let (Ω,A,µ,T) be an ergodic dynamical system on a Lebesgue probability space, (a n ) n N R + be an increasing sequence such that a n ր and an 0 as n. n There exists a dense (for the pseudo-metric of the measure of the symmetric difference) G δ class of sets A A having the property that for every probability ν on R, there exists a subsequence (n k ) k N satisfying S nk (l A µ(a)) D k ν. Here, we answer the question of the existence of a similar result in the non-ergodic case. Assume now that (Ω,A,µ,T) is not ergodic. Let (a n ) n N R + be an increasing sequence satisfying a n ր and an 0 as n. Denote by I the σ-algebra of the n invariant sets and (µ x ) x χ the ergodic components of the measure µ. If there exist a set A A, a probability measure ν on R and a sequence (n k ) k N such that then E(l A I) = µ(a) almost surely. S nk (l A µ(a)) D k ν, Indeed, if there exists x χ such that µ x (A) µ(a) = c > 0, then by Birkhoff s Ergodic Theorem, n S n(l A µ(a)) c n µ x -almost surely. Therefore and we have a contradiction. S nk (l A µ(a)) k + So, to find a set which satisfies the conclusion of Theorem, we have to consider the sets A such that E(l A I) is almost surely constant. The class of such sets is not, in general, dense in A. So, in the non-ergodic case, we cannot expect the result of genericity. Nevertheless, in the non-ergodic case, one can show the( existence of an arbitrarily ) small set A A such that the sequence of the distributions of a n S n (l A µ(a)) is dense n N in the set of probability measures on R. We prove the following result. 2

3 Theorem 2 Let (Ω,A,µ,T) be an aperiodic dynamical system on a Lebesgue probability space and (a n ) n N R + be an increasing sequence such that a n ր and an n 0 as n. For all ε > 0, there exists a set A A with µ(a) < ε such that for every probability measure ν on R, there exists a sequence (n k ) k N such that S nk (l A µ(a)) D k ν. The rest of the paper is devoted to the proof of Theorem 2. Note that the proof that we propose is constructive. 2 Proof Let (Ω,A,µ,T) be an aperiodic dynamical system and (a n ) n N R + be an increasing sequence such that a n ր and an 0 as n which are fixed for all the sequel. n 2. An equivalent statement Let M be the set of all probability measures on R and M 0 be the set of all probability measures on R which have zero-mean. We denote by d the Lévy metric on M. For all µ and ν in M with distribution functions F and G, d(µ,ν) = inf{ε > 0 : G(t ε) ε F(t) G(t+ε)+ε, t R}. The space (M, d) is a complete separable metric space and convergence with respect to d is equivalent to weak convergence of distributions (see Dudley [4], pages ). If X : Ω R is a random variable, we denote by L Ω (X) the distribution of X on R. Using the separability of the set M 0 which is dense in M, we can prove that the next theorem is equivalent to Theorem 2. Theorem 3 For every ε > 0 and for every sequence (ν k ) k N in M 0, there exist a set A A, with µ(a) < ε, and a sequence (n k ) k N such that d(l Ω ( S nk (l A µ(a))),ν k ) k 0. Proposition 2. Theorem 3 and Theorem 2 are equivalent. Let M be a countable and dense subset of M 0. We can find a sequence (ν k ) k N such that for all η M, there exists an infinite set K η N verifying that for all k K η, ν k = η. Let A be the set and (n k ) k N be the sequence associated to the sequence (ν k ) k N as in Theorem 3. 3

4 For each ν M, there exists an increasing sequence (k j ) j N such that ν kj = ν for all j N. By Theorem 3, By classical argument, Theorem 2 follows. d(l Ω ( j S nkj (l A µ(a))),ν) j 0. The fact that Theorem 2 implies Theorem 3 is clear. Now to prove Theorem 3, we will construct explicitly the set A. To do that, we will use the four following lemmas. 2.2 Auxiliary results Let ν be a probability on R. For B B(R) with ν(b) > 0, ν B denotes the probability on R defined by ν B (A) = ν(b) ν(a B). For x R, ν x denotes the probability on R defined by ν x (B) = ν({xb/b B}). Lemma 2.2 Some properties of the Lévy metric: (i) For each probability ν on R, for all Borel sets B, d(ν B,ν) ν(r\b). (ii) For all probabilities ν and η on R, for all x, d(ν x,η x ) d(ν,η). (iii) For all probability ν on R, for all measurable functions f and g from Ω to R, where δ 0 is the Dirac measure at 0. d( L Ω (f +g),ν) ( L Ω (f),ν)+d( L Ω (g),δ 0 ) (iv) For all probability ν on R, d(ν,δ 0 ) A if and only if ν((, A)) A and ν((a, )) A. The proof is left to the reader. Lemma 2.3 For all probability ν on R, for all ε > 0, there exists C 0 and n 0 N, for all C C 0 and n n 0, there exists a probability η on R with support S [ a n C,a n C] Z such that for all i S, η({i}) Q, d(η an,ν) ε and E(η) := xdη(x) = 0. This Lemma is a consequence of Lemma 3.3 in [5] (which has a constructive proof) and of the fact that for all probability measure on Z with finite support, we can find a probability on Z with same support which is arbitrarily close to the first one (with respect to d) and takes values in Q. The following lemma is classical, we do not give a proof. 4

5 Lemma 2.4 Let (Ω,A,µ) be a Lebesgue probability space and ν be a probability on R. Then, there exists a measurable random variable X : Ω R, such that L Ω (X) = ν. Recall that a set F A is the base of a Rokhlin tower of height n if the sets F,TF,...,T n F are pairwise disjoint. Lemma 2.5 For all n and for all ε > 0, there exists a set F A such that {F,...,T n F} is a Rokhlin tower of measure greater than ε and the sojourn time in the junk set J = Ω\( i=0 n Ti F) is almost surely, i.e. for a.e. x J, Tx F. This can be view as a consequence of Alpern s theorem [], by constructing a Rokhlin castle with two towers of height n and n+ and the base of the second tower of measure less than ε. 2.3 Proof of Theorem 3 Let the sequence (ν k ) k in M 0 and the constant ε (0,) be fixed. Let (ε k ) k be a decreasing sequence of positive reals such that k ε k < ε and k kε k <. Theorem 3 is a consequence of the following proposition, which is proved in the next section. Proposition 2.6 There exist a sequence of pairwise disjoint sets A k A and a sequence of integers (n k ) k such that, (i) µ(a ) ε and for all k >, µ(a k ) an k n k ε k ; (ii) for all k, d(l Ω ( S nk (l Ak µ(a k ))),ν k ) ε k ; (iii) for all k and for all j > k, d(l Ω ( a nj S nj (l Ak µ(a k ))),δ 0 ) ε j. We admit the proposition for the end of the proof. Set A = k A k. Then by (i), µ(a) = k µ(a k ) k ε k ε. 5

6 By Proposition 2.6 and by (iii) and (iv) of Lemma 2.2, for all j, d(l Ω ( a nj S nj (l A µ(a))),ν j ) j d(l Ω ( S nj (l Ai µ(a i ))),δ 0 ) a nj i= +d(l Ω ( S nj (l Aj µ(a j ))),ν j ) a nj + d(l Ω ( S nj (l Ai µ(a i ))),δ 0 ) a nj i=j+ (j )ε j +ε j + i=j+ n j a nj µ(a i ) i j iε i which goes to 0 when j goes to. Thus Theorem 3 is proved. 2.4 Proof of Proposition 2.6 We give an explicit construction of the sets A k. We begin by the construction of the set A. Step : the set A The goal is to find a set A and an integer n such that d(l Ω ( a n S n (l A µ(a ))),ν ) ε. But we also want that the set A becomes negligible for thepartial sums oflength n k, k 2 (condition (iii)). There are several steps. First, we will define a set A, which satisfies (ii) and (iii) for j = 2. Then, we will modify this set, step by step, to have (iii) for all j > 2. The set A, We consider the probability ν, the constant ε and we set α := ε 8. Applying Lemma 2.3toν andα,wegettwoconstantsc(ν,α )andn(ν,α )andwechoosec := C(ν,α ) and n n(ν,α ) such that d n α where d := a n C +. () We get a corresponding centered probability η (given by Lemma 2.3) with support in { d +,...,d } such that d(η an,ν ) α. Since for all i, η ({i}) Q, there exist q N and q (i) N, i =...,2d, such that η ({i d }) = q(i), for all i =,...,2d. q 6

7 Now, weconsidertheprobabilityν 2 andε 2. Wedefineα 2 := an 2n ε 2. ApplyingLemma2.3 to ν 2 and α 2, we get two constants C(ν 2,α 2 ) and n(ν 2,α 2 ). Set C 2 := max{c(ν 2,α 2 ),C } and let n 2 n(ν 2,α 2 ) be a multiple of q n such that q n a n2 α 2. (2) By Lemma 2.5, we can consider a set F A such that {F,TF,...,T n2 F } is a Rokhlin tower of height n 2, with the sojourn time in the junk set almost surely equal to and the measure of the junk set smaller than γ := min{ an 2 n 2 α 2,α }. Write F l := T ln F, l = 0,...,p. We thus have p := n 2 n towers {F l,...,tln F l} of height n. Notice that by definition of n 2, p is a multiple of q. By Lemma 2.4, let h be a measurable function from F to Z such that L F (h ) = η and denote by g the positive function equal to h +d. Let A F,i := g ({i}), i =,...,2d. Now, for all i =,...,2d, let {A F,i,,...,A F } be a partition of the set A,i,q (i) F,i into sets of measure q µ(f ). We thus have a partition of F into {A F,,,...,A F,,q (),A F,2,,...,A F,2,q (2),...,A F,2d,,...,A F,2d,q (2d }. ) By induction, we define partitions of F l for l =,...,p by setting T n A F l,i,j+ if j q (i),i,j := T n A F l,i+, if j = q (i) and i < 2d. T n A F l,, if j = q (2d ) and i = 2d For all l = 0,...,p and for all i =,...,2d, we set and,i := := q (i) j= 2d i i=k=0,i,j T k,i. Remark that for any l {0,...,p }, for any x F l and for any i {,...,2d }, Now, we define the set A, as follows x,i if and only if S n (l AF l )(x) = i. A, := p 7 l=0.

8 Remark that for any x F, S n q (l A, )(x) = d q and since p is a multiple of q, n S n2 (l A, )(x) = d p. Moreover, for each k {,..., 2 n q }, for any x T kn q F, we also have S n q (l A, )(x) = d q. Lemma 2.7 (i) µ(a, ) α ; (ii) A, p j=0 2d i=0 T i+jn F ; (iii) d(l Ω ( a n S n (l A, µ(a, ))),ν ) ε ; (iv) d(l Ω ( a n2 S n2 (l A, µ(a, ))),δ 0 ) α 2. For each l = 0,...,p, µ( ) = E F (g )µ(f ) = d µ(f ). Therefore, by (), µ(a, ) = p l=0 By construction, (ii) is clear. Let Ω := p l=0 µ( ) = p d µ(f ) p n 2 d = d n α. n 2d i=0 T i F l. We have and since E F (g ) = d, by centering, Now, since γ α, L Ω (S n (l A, )) = L F (g ) L Ω (S n (l A, µ(a, ))) = L F (h ). µ(ω ) = p (n 2d )µ(f ) (n 2 2p d ) ( γ ) n 2 = γ 2d n 3α. Thus, by Lemma 2.2 (i), and by Lemma 2.2 (ii), d(l Ω (S n (l A, µ(a, ))),L F (h )) 3α. d(l Ω ( a n S n (l A, µ(a, ))),L F ( h a n )) 3α. 8

9 We infer, by triangular inequality, that and (iii) is proved. d(l Ω ( a n S n (l A, µ(a, ))),ν ) ε Recall that n 2 is a multiple of n q and, by definition of A,, S n q (l A, )(x) = d q whenever x belongs to one of the T kn q n F for k = 0,..., 2 n q. Since the sojourn time in the junk set is, we infer that for any x Ω, Using µ(a, ) = p d µ(f ), we get (p q )d S n2 (l A, )(x) (p +q )d. S n2 (l A, µ(a, )) p d n 2 µ(f ) +q d p d γ +q d. Thus, since γ an 2 n 2 α 2 and by (2), we have a n2 S n2 (l A, µ(a, )) d n α 2 + d n q n a n2 2α α 2 α 2 and (iv) follows from application of Lemma 2.2 (iv). Of course, the set A, is not defined well enough to be negligible for higher partial sums. So, we need to modify a small part of A,. Thus we introduce a sequense of sets A,k, k 2. The set A, can be considered as a first version of the set A and the A,k, k 2 are the adjustments. The sets A,k, k 2 We shall give here the general algorithm to deduce the set A,k from A,k. To do that, we need first to define the entire sequence (n k ) k. By induction, we define the sequences (α k ) k 2, (C k ) k 2, (n k ) k 2, (q k ) k 2 as follows. We consider the probability ν k and ε k. We define α k := an k 2n k ε k. Applying Lemma 2.3 to ν k and α k, we get two constants C(ν k,α k ) and n(ν k,α k ). Set C k := max{c(ν k,α k ),C k } and let n k n(ν k,α k ) be a multiple of q k n k such that and d k n k α k where d k := C k +. (3) q k n k α k. (4) By Lemma 2.3, we get a corresponding centered probability η k on { d k +,...,d k } such that d(η kank,ν k ) α k. There exist q k N and q (i) k N, i =...,2d k, such that η k ({i d k }) = q(i) k q k. 9

10 For k, we also set p k := n k+ n k N and β k := α k α k+. Thus, for all k, We define the sequence (γ k ) k by γ k := min β j α k. (5) j k { βk+, a } n k+ α k+. (6) 2p k+ n k+ Further, for all k, by application of Lemma 2.5, we obtain a set F k A such that {F k,tf k,...,t n k+ F k } is a Rokhlin tower of height n k+ and the junk set J k := Ω\ n k+ i=0 T i F k is a set with sojourn time and µ(j k ) γ k. Remark that α 2, n 2 and γ 2 have been previously defined but they respect this new definition. We also introduce the sequence of sets F k defined by induction in the following way: F := F and F k := T n(x) x. x F k where for all x in F k, n(x) := inf{n 0/T n x F k } is the time of the first visit in F k. Lemma 2.8 There exists a sequence of measurable sets (A,k ) k such that (i) µ(a,k A,k ) β k ; (ii) for all x Ω, S n (l A,k )(x) 2d ; (iii) for all x F k, for all i = 0,...,p k, S nk (l A,k T in k )(x) = n k n d ; (iv) d(l Ω ( S nk+ (l A,k µ(a,k )),δ 0 ) α k+. + We prove the lemma by induction. The set A, is already defined. Now, for a fixed k, we are going to explain how to deduce the set A,k from A,k. For x F k and i = 0,...,p k, let ρ i (x) := (i+)n k j=in k l A,k T j (x) = S nk (l A,k T ink ). Byhypothesis, forallx F k, ρ n 0(x) = S nk (l A,k )(x) = p k k n d = n k n d butfori > 0 it can be different. The differences appear when the orbit of the point x meets the junk 0

11 set J k. Nevertheless, by definition of the Rokhlin tower (see Lemma 2.5), it can meet J k only one time in every n k consecutive iterates by T. So we have, (p k ) n k n d ρ i (x) (p k +) n k n d. To summarize, for x F k and i = 0,...,p k, ρ i (x) = n k n d +j with j { n k n d,..., n k n d }. We define a set B i (x) as follows. If j 0, B i (x) =. If j < 0, let B i (x) be a set composed by j points from the set {T in k x,...,t (i+)n k x}\a,k, in such a way that every n -consecutive points in {T in k x,...,t (i+)n k x} meet A,k B i (x) at most 2d times (it is possible because otherwise, ρ i (x) > n k n 2d j n k n d ). We define a set C i (x) as follows. If j 0, C i (x) =. If j > 0, let C i (x) be the set composed by the j first points of {T in k x,...} A,k. and Let B := x F k p k i=0 B i (x), C := x F k A,k := (A,k \C) B. p k i=0 C i (x) Since the orbit of a point x can only meet J k one time every n k and using (6), we have µ(a,k A,k ) 2p k µ(j k ) 2p k γ k β k. Remark that (ii) and (iii) are guaranteed by construction of A,k. Further for all x F k, we have S nk+ (l A,k )(x) = p k n k n d = n k+ n d. We deduce that µ(a,k ) n k+ n d µ(f k ) µ(j k) and (p k ) n k n d S nk+ (l A,k ) (p k +) n k n d. Then, S nk+ (l A,k µ(a,k )) n k d +(+ n k+ d )µ(j k ) n n and by (6) and (4), S nk+ (l A,k µ(a,k )) n k d +( + d )α k+ α k n n k+ n By Lemma 2.2 (iv), we get (iv).

12 The set A We can now define the set A A as A := lim k A,k which is well defined because the sequence (µ(a,k A,k+ )) k is summable. Lemma 2.9 (i) µ(a ) 2α ; (ii) S n (l A ) 2d ; (iii) d(l Ω ( a n S n (l A µ(a ))),ν ) 2ε ; (iv) For all k 2, d(l Ω ( S nk (l A µ(a ))),δ 0 ) ε k. For all k, we have µ(a A,k ) j=k+ µ(a,j A,j ) and then µ(a ) µ(a, )+µ(a A, ) 2α. Assertion (ii) comes from Lemma 2.8 (ii). Further (7) and Lemma 2.2 (iv) imply that for all n, j=k+ β j α k+ (7) d(l Ω ( a n (S n (l A,k µ(a,k )) S n (l A µ(a )))),δ 0 ) n a n α k+. Using Lemma 2.2 (iii), we can deduce (ii) from Lemma 2.7 (iii) and (iii) from Lemma 2.8 (iv). Step 2: The set A 2 The set A 2,2 We consider F 2 A and we will almost repeat what we did to find the set A,, working with n 2,q 2,p 2,d 2 instead of n,q,p,d. The difference comes to the fact that we want A A 2 =. Recall that η 2 is the probability measure with support in Z given by Lemma 2.3 applied to ν 2 and α 2 and with constants C 2 and n 2. Let h 2 be a function from F 2 to Z given by Lemma 2.4 such that L G (h 2 ) = η 2 and call g 2 the positive function equal to h 2 +d 2. Let A F2,i := g 2 ({i}), i =,...,2d 2 2

13 and for all i =,...,2d 2, let {A F2,i,,...,A F2 } be a partition of the set A,i,q (i) F2,i into sets 2 of measure q 2 µ(f 2 ). We thus have a partition of F 2 into {A F2,,,...,A F2,,q () 2,A F2,2,,...,A F2,2,q (2) 2,...,A F2,2d,,...,A F2,2d,q (2d }. ) 2 By induction, we deduce partitions of F2 l = Tln 2 F 2 for l =,...,p 2. We set T n 2 A F l 2,i,j+ if j q (i) 2 2,i,j := T n 2 A F l 2,i+, if j = q (i) 2 and i < 2d 2. T n 2 A F l 2,, if j = q (2d ) 2 and i = 2d 2 For all l = 0,...,p 2 and for all i =,...,2d 2, we set 2,i := q 2 (i) j= 2,i,j. Because we want disjointness, we cannot define 2 as 2d 2 i i= k=0 Tk 2,i. So, for each l {0,...,p 2 }, for each x F2 l, if x A F2 l,i, we denote by D l (x) the set composed by the i first elements of {x,tx,...}\a and we set D l := D l (x). Since A contains at most 2d points in each orbit of size n and since d 2 d, x F l 2 We define A 2,2 as D l A 2,2 := 4d 2 i=0 p 2 l=0 T i F l 2. D l. Lemma 2.0 (i) µ(a 2,2 ) α 2 ; (ii) S n2 (l A2,2 ) 2d 2 ; (iii) d(l Ω ( a n2 S n2 (l A2,2 µ(a 2,2 ))),ν 2 ) ε 2 ; (iv) d(l Ω ( a n3 S n3 (l A2,2 µ(a 2,2 ))),δ 0 ) α 3. The proof follows the one of Lemma 2.7 and is left to the reader. 3

14 The sets A 2,k, k 3 Nowwedefineasequence A 2,k,k 3usingthesametechniques asbeforeandpreserving the fact that for all k, A,k contains at most 2d 2 points in each orbit of length n 2. Then the A 2,k satisfy µ(a 2,k A 2,k ) β k and d(l Ω ( S nk+ (l A2,k µ(a 2,k ))),δ 0 ) α k+. + The set A 2 The set A 2 := lim A 2,k k is well defined, disjoint of A and satisfies the following lemma. Lemma 2. (i) µ(a 2 ) 2α 2 ; (ii) S n2 (l A2 ) 2d 2 ; (iii) d(l Ω ( a n2 S n2 (l A2 µ(a 2 ))),ν 2 ) 2ε 2 ; (iv) For all k 3, d(l Ω ( S nk (l A2 µ(a 2 ))),δ 0 ) ε k. The proof follows the one of Lemma 2.9. Step k: the set A k, k 3 It is now clear that, by induction, we can find sets A k disjoint of the sets A i, i < k, satisfying Proposition 2.6. References [] Steve Alpern. Generic properties of measure preserving homeomorphisms. In Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 978), volume 729 of Lecture Notes in Math., pages Springer, Berlin, 979. [2] Robert Burton and Manfred Denker. On the central limit theorem for dynamical systems. Trans. Amer. Math. Soc., 302(2):75 726, 987. [3] Andrés del Junco and Joseph Rosenblatt. Counterexamples in ergodic theory and number theory. Math. Ann., 245(3):85 97,

15 [4] R. M. Dudley. Real analysis and probability, volume 74 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, Revised reprint of the 989 original. [5] Olivier Durieu and Dalibor Volný. On sums of indicator functions in dynamical systems. to appear in Ergodic Theory and Dynamical Systems. [6] Pierre Liardet and Dalibor Volný. Sums of continuous and differentiable functions in dynamical systems. Israel J. Math., 98:29 60, 997. [7] Dalibor Volný. On limit theorems and category for dynamical systems. Yokohama Math. J., 38():29 35,