Central Limit Theorems. in Ergodic Theory

Transcription

1 Cetral Limit Theorems i Ergodic Theory Attila Herczegh Thesis for M.Sc. degree i Mathematics Advisor: Vilmos Prokaj Eötvös Lorád Uiversity Faculty of Sciece, Eötvös Lorád Uiversity, Budapest Budapest, 2009.

2 Cotets Ackowledgemets iii Itroductio. Basic defiitios ad examples Etropy Equipartitios 6 2. Cetral limit theorem Weak ivariace priciple Examples CLT-property property ii

3 Ackowledgemets I would like to express my gratitude to my advisor i Budapest, Vilmos Prokaj, ad my former supervisor, Karma Dajai, for all the advice, support ad ecouragemet they gave me throughout the time I spet workig o this thesis. It has bee a pleasure to work with both of them. I also thak for my uiversities, the Eötvös Lorád Uiversity i Budapest ad the VU Uiversity Amsterdam i Amsterdam, to make everythig possible. iii

4 Chapter Itroductio. Basic defiitios ad examples First of all, let us start with a little itroductio about ergodic theory. Ergodic theory is the study of the log term average behaviour of systems evolvig i time. As a startig poit we have a probability space (X, B, µ). The collectio of all states of the system form a space X ad the evolutio, i our case, is represeted by a measurable trasformatio T : X X, so that T A B for all A B. I most cases we would like our trasformatio to be measure preservig ad ergodic, i.e.: Defiitio.. Let (X, B, µ) be a probability space ad T : X X measurable. The map T is said to be measure preservig with respect to µ if µ(t A) = µ(a) for all A B. We call the quadruple (X, B, µ, T ) a measure preservig or dyamical system. Defiitio.2. Let T be a measure preservig trasformatio o a probability space (X, B, µ). T is called ergodic with respect to µ if for every measurable set A satisfyig A = T A, we have µ(a) = 0 or. Example.. Cosider ([0, ), B, µ), where B is the Borel σ-algebra ad µ is the Gauss measure give by the desity log 2 +x with respect to the Lebesgue measure. Let T be the Gauss trasformatio give by T (x) = x (mod ). It is well kow that T preserves the Gauss measure. Moreover, T is ergodic with respect to µ. Example.2. Cosider ([0, ), B, λ), where B is the Lebesgue σ-algebra ad λ is the Lebesgue-measure. Let T : [0, ) [0, ) be give by T x = rx mod, where r is a positive iteger. The T is ergodic with respect to λ.

5 CHAPTER. INTRODUCTION 2 Example.3. Cosider ([0, ), B, λ) as above. Let > a o-iteger, ad cosider T : [0, ) [0, ) give by T x = x mod. The T is ergodic with respect to λ, i.e. if T A = A, the λ(a) = 0 or. The followig lemma provides, for example i the cases metioed above, a useful tool to verify ergodicity of a measure preservig trasformatio defied o ([0, ), B, µ), where µ is equivalet to the Lebesgue measure. Lemma.. (Kopp s lemma) If B is a Lebesgue-set ad C is a class of subitervals of [0, ) satisfyig (a) every ope subitervals of [0, ) is at most a coutable uio of disjoit elemets from C, (b) A C, λ(a B) γλ(a), where γ > 0 is idepedet of A, the λ(b) =. A importat theorem is the Ergodic Theorem also kow as Birkhoff s Ergodic Theorem, which is i fact a geeralizatio of the Strog Law of Large Numbers. The theorem goes as follows: Theorem.. (Ergodic Theorem) Let (X, B, µ) be a probability space ad let T : X X be a measure preservig trasformatio. The f L (X, B, µ), lim f(t i x) = f (x) i=0 exists µ-a.e. x X, is T -ivariat ad X fdµ = X f dµ. If moreover T is ergodic, the f is a costat a.e. ad f = X fdµ. This is widely used theorem, for example, it is crucial i the proof of the Shao-McMilla-Breima theorem which we will state later. Usig the Ergodic Theorem oe ca give aother characterizatio of ergodicity: Corollary.. Let (X, B, µ) be a probability space ad T : X X a measure preservig trasformatio. The T is ergodic if ad oly if for all A, B B oe has lim µ ( T i B A ) = µ(a)µ(b). i=0 This corollary gives a ew defiitio for ergodicity, amely, the asymptotic average idepedece. We ca defie other otios like this which are stroger tha ergodicity, called mixig properties: Defiitio.3. Let (X, B, µ) be a probability space ad T : X X a measure preservig trasformatio. The, (i) T is weakly mixig if for all A, B B oe has lim ( µ T i B A ) µ(a)µ(b) = 0. i=0

6 CHAPTER. INTRODUCTION 3 (ii) T is strogly mixig if for all A, B B oe has lim µ ( T i B A ) = µ(a)µ(b). It is ot hard to see that strogly mixig implies weakly mixig ad weakly mixig implies ergodicity. This follows from the simple fact that if {a } such that lim a = 0 the lim a = 0 ad hece lim a = 0. The followig propositio helps us check whether a trasformatio is mixig: Propositio.. Let (X, B, µ) be a probability space ad T : X X a measure preservig trasformatio. Let C be a geeratig semi-algebra of B. The, (a) If for all A, B C oe has lim µ ( T i B A ) µ(a)µ(b) = 0, i=0 the T is weakly mixig. (b) If for all A, B C oe has the T is strogly mixig. lim µ ( T i B A ) = µ(a)µ(b), Example.4. (Beroulli shifts) Let X = {0,,..., k } N, B geerated by the cyliders. Let p = (p 0, p,..., p k ) be a positive probability vector. We defie the measure µ o B by specifyig it o the cylider sets as follows: µ({x: x 0 = a 0,..., x = a }) = p a0 p a p a. Let T : X X be defied by T x = y where y = x +. This map T, called the left shift, is measure preservig ad eve strogly mixig with respect to µ. The measure preservigess follows easily from the fact that k T {x: x 0 = a 0,..., x = a } = {x: x 0 = j, x = a 0,..., x + = a }. j=0 The cylider sets form a semi-algebra, so we ca use Propositio.. Take A, B cyliders: A = {x: x 0 = a 0, x = a,..., x k = a k } ad B = {x: x 0 = b 0, x = b,..., x m = b m }. Now if we take m +, the T A ad B specify differet coordiates, thus µ(t A B) = µ({x: x 0 = b 0,..., x m = b m, x = a 0,..., x +k = a k }) = = µ({x: x 0 = b 0,..., x m = b m })µ({x: x = a 0,..., x +k = a k }) = = µ(b)µ(t A) = µ(a)µ(b), which implies that T is strogly mixig.

7 CHAPTER. INTRODUCTION 4 Example.5. (Markov shifts) Let (X, B, T ) be as above. We defie a measure ν o B by specifyig it o the cylider sets as follows: Let P = (p ij ) be a stochastic k k matrix, ad q = (q 0, q,..., q k ) a positive probability vector such that qp = q. Defie ν the followig way: ν({x: x 0 = a 0,..., x = a }) = q a0 p a0a p a a. Just as i the previous example, oe ca easily see that T is measure preservig with respect to ν. Note that the Beroulli shifts are Markov shifts with q = p ad with P = (p 0, p,..., p k ) where deotes the the vector for which all the k coordiates are..2 Etropy There is a very importat otio i ergodic theory called etropy. To defie it, we eed a few steps. First, let α be a fiite or coutable partitio, i.e. X is a disjoit uio (up to measure 0) of A α. We defie the etropy of the partitio α by H(α) = H µ (α) := A α µ(a) log µ(a). Here ad from ow o everywhere log represets logarithm with base 2. If α ad are partitios, the we defie α := {A B : A α, B } ad uder T α we cosider the partitio T α = {T A: A α}. Now cosider the partitio i=0 T i α, whose atoms, i.e. the members of the partitio, are of the form A i0 T A i T ( ) A i. The the followig propositio ca be prove: Propositio.2. Let α be a fiite or a coutable partitio of the dyamical system (X, B, µ, T ) with T measure preservig trasformatio. Assume that H(α) <. The lim H( i=0 T i α) exists. The proof is based o the fact that the above sequece is subadditive, as it ca be show that H(α ) H(α) + H() ad it is obvious that if T is µ-ivariat, the H(T α) = H(α). This propositio allows to defie the followig: Defiitio.4. The etropy of the measure preservig trasformatio T with respect to the partitio α is give by h(α, T ) = h µ (α, T ) := lim H( T i α). i=0

8 CHAPTER. INTRODUCTION 5 For each x X, let us itroduce the otatio α(x) for the elemet of α to which x belogs. We close the itroductio by statig oe of the most importat theorems of ergodic theory, which we are goig to use quite ofte: Theorem.2. (The Shao-McMilla-Breima Theorem) Suppose T is a ergodic measure preservig trasformatio o a probability space (X, B, µ), ad let α be a coutable partitio with H(α) <. Deote α = i=0 T i α, the lim log µ(α (x)) = h(α, T ) µ a.e. For proofs ad more detailed itroductio, see([2]).

9 Chapter 2 Equipartitios I this chapter we will ivestigate the rate at which the digits of oe umbertheoretic expasio determie those of aother. First of all, let us defie what a umber-theoretic expasio is: Defiitio 2.. A surjective map T : [0,) [0,) is called a umber-theoretic fibered map (NTFM) if it satisfies the followig coditios: (a) there exists a fiite or coutable partitio of itervals P = {P i : i D} such that T restricted to each atom of P (cylider set of order 0) is mootoe, cotiuous ad ijective, (b) T is ergodic with respect to Lebesgue measure λ, ad there exists a T - ivariat probability measure µ equivalet to λ with bouded desity. (Both dµ dλ are bouded.) ad dλ dµ Iteratios of T geerate expasios of poits x [0, ) with digits i D. We refer to the resultig expasio as the T -expasio of x. Throughout the chapter we are goig to assume the followigs: let T ad S be umber-theoretic fibered maps o [0,) with probability measures µ ad µ 2 respectively, each boudedly equivalet to Lebesgue measure ad with geeratig partitios (cyliders of order 0) P ad Q respectively. Deote by P ad Q the iterval partitios of [0,) ito cylider sets of order, amely, P = i=0 T i P ad Q = i=0 S i Q. Deote by P (x) the elemet of P cotaiig x (similarly for Q (x)), ad itroduce m(, x) = sup{m 0 P (x) Q m (x)}. Suppose that H µ (P ) ad H µ2 (Q) are fiite ad = h µ (T ) ad = h µ2 (S) are positive. 2. Cetral limit theorem Our startig poit i this sectio is the followig theorem by Dajai ad Fieldsteel[3]: 6

10 CHAPTER 2. EQUIPARTITIONS 7 Theorem 2.. m(, x) lim = a.e. The same holds true if we look at the smallest k for which P k (x) is still cotaied i Q m (x). So let us defie k(m, x) = if{k 0 P k (x) Q m (x)}. The uder the same assumptios, the followig is true: Theorem 2.2. k(m, x) lim m = a.e. Proof. Let us take x such that the covergece i Theorem 2. holds. The for all m > m(, x) we ca fid N such that m(, x) < m m( +, x). From the defiitio of m(, x), this meas that P + (x) Q m(+,x) (x) Q m (x) ad that P (x) Q m (x), i.e. k(m, x) = +. Thus + m( +, x) + k(m, x) = m m < + m(, x). Here both the first ad the last terms coverge to from the previous theorem, which meas that k(m,x) m coverges ad the limit is what we wated. Sice the covergece i Theorem 2. holds almost everywhere, we have completed the proof. I the sequel the followig two properties are goig to play a importat role: Property 2.. We say that the triplet (P, µ, T ) or i short the trasformatio T satisfies the 0-property if log µ (P (x)) 0 almost everywhere. Property 2.2. We say that the triplet (Q, µ 2, S) or i short the trasformatio S satisfies the CLT-property if ( ) log lim µ µ2 (Q (x)) 2 σ u = for every u R ad for some σ > 0. u 2π e x2 2 dx,

11 CHAPTER 2. EQUIPARTITIONS 8 Remark 2.. Notice that both properties ca be stated with the Lebesgue measure sice we oly look at measures equivalet to it. I the case of the CLTproperty, we mea that log µ 2 (Q (x)) ca be replaced with log λ(q (x)). Lemma 2.. If the trasformatio T satisfies the 0-property, the ( ) λ(qm (x)) log = o ( m ) i probability (2.) λ(p k(m,x) (x)) k(m,x) Proof. We kow that lim m m = > 0 from Theorem 2.2. From the zero property we also kow that for large ad for most of the x poits log(λ(p + (x))) log(λ(p (x))) + o( ). We combie these two properties to get the result. Ideed, sice by defiitio Q m P k(m,x), we see that the left had side of (2.) is o egative. So we oly have to prove that for ay positive ε there is a m 0 such that for m > m 0 λ ({ x [0, ]: log (λ(q m (x))) log (λ(p k(m,x) (x))) > ε m }) < ε Fix ε > 0 ad put A = { x [0, ]: log(λ(p (x))) + > } 2 ε. By the zero property there is a 0 such that λ( 0 A ) ε. Put { B = x: m >, k(m, x) < 2 m } < 2 Sice k(m,x) m has a fiite positive limit m > 2 0 such that λ (B m ) > ε. Now if x / 0 A the for > 0 we have that for almost every x, we ca fid log(λ(p (x))) log(λ(p (x))) + ε So if m > m ad x B m \ 0 A the Hece d(x, Q m (x)) λ(p k(m,x) (x)) 2 +ε k(m,x) λ(p k(m,x) (x)) 2 +ε 2 m λ(p k(m,x) (x)) log (λ(q m (x))) log ( λ(p k(m,x) (x)) ) m log (λ(q m (x))) log(d(x, Q m (x))) + m + ε 2. (2.2)

12 CHAPTER 2. EQUIPARTITIONS 9 For each m ad I Q m let I I a cocetric iterval such that λ(i ) = ( ε)λ(i). The λ( I Qm I ) = ε ad for each x I Qm I the right had side of (2.2) is ot greater tha log ( 2 ) ε + + ε m 2. Now we ca put the above pieces together. Let m 0 > m so large that ( log 2 ) ε + + ε 2 m0 < ε 3. For m > m 0 ad x (B m I Q m I ) \ 0 A all the above estimatio are valid, hece ({ λ x [0, ]: log (λ(q m (x))) log ( λ(p k(m,x) (x)) ) }) > ε 3 m < 3ε. Sice ε > 0 was arbitrary, the proof is complete. Now we ca prove the followig theorem which says that the speed of covergece i Theorem 2.2 is i fact of order m : Theorem 2.3. Let us suppose that the trasformatio S satisfies the CLTproperty ad that T satisfies the 0-property. The the followig holds: where σ = measure µ 2. σ k(m, x) m σ m N (0, ), ad is the covergece i law with respect to the probability Proof. We ca divide the expressio above ito four parts the followig way: = log λ(p k(m,x)(x)) + k(m, x) σ m k(m, x) m σ m = + log λ(q m(x)) + log µ 2 (Q m (x)) σ m + log λ(p k(m,x)(x)) + log λ(q m (x)) σ m + + log µ 2(Q m (x)) m σ m. Here the first term goes to 0 for almost every x because of the coditios made o the trasformatio T. To see this, let us take x [0, ) such that the covergece i k(m,x) m is satisfied. It is eough to see that for such a x the first term goes to 0. We ca prove this easily: log λ(p k(m,x) (x)) + k(m, x) σ m =

13 CHAPTER 2. EQUIPARTITIONS 0 k(m, x) = σ m log λ(p k(m,x) (x)) + k(m, x) k(m, x), k(m,x) ad sice the multiplier m σ < ad the rest is a subsequece of the opposite of the sequece appearig i the assumptio of the 0-property sice k(m, x) as m, the whole thig goes to 0. λ(q The secod term equals to σ m log m(x)) λ(p k(m,x) (x)) ad thus goes to 0 i probability by Lemma 2.. The third term will go to 0 for every x because of the equivalecy of the measures λ ad µ 2. That is to say there K ad K 2 positive ad fiite costats such that for every Borel-set B: K µ 2 (B) λ(b) K 2 µ 2 (B). Hece σ log K = log K log µ 2 (Q m (x)) + log µ 2 (Q m (x)) σ m σ m log λ(q m(x)) + log µ 2 (Q m (x)) σ m log K 2 log µ 2 (Q m (x)) + log µ 2 (Q m (x)) = log K 2. σ m σ m Here both the first ad the last expressio go to 0 for every x ad thus so does the the third term above. So all is left is the fourth term, ad that oe has a limitig stadard ormal distributio sice we have assumed that for the trasformatio S ad we have chose σ = theorem. σ. This meas that altogether k(m, x) satisfies a cetral limit Remark 2.2. From ow o there will be times whe we would like to show that somethig coverges i probability usig that for some other expressio we already kow this. Here is how we are goig to proceed i these cases. We would like to use the expressio - eve i coectio with a sequece of radom variables that oly coverges i probability - that we take a poit where the covergece holds. By this we are goig to mea the followig. It is wellkow that a sequece of radom variables coverges i probability if ad oly if for every subsequece there is a sub-subsequece which coverges almost surely. So if we wat to show that a sequece coverges i probability, we ca take a subsequece ad we oly eed to show that there is a sub-subsequece which coverges almost surely. Ad vice versa, if we have a sequece that coverges i probability, the for every subsequece of it, we ca take a sub-subsequece that coverges almost everywhere. So whe we say for a sequece of radom variables which coverges i probability that let us take a poit where the covergece holds, we mea that there has bee a subsequece for which we ca take a sub-subsequece which coverges almost everywhere ad thus we take a poit where this almost sure covergece holds. But we do ot wat to bother with subsequeces ad sub-subsequeces which would cause some problems with the otatios. However, it is very importat to bear i mid that this is what we mea by that.

14 CHAPTER 2. EQUIPARTITIONS 2.2 Weak ivariace priciple The cetral limit theorem ca be improved to what is called the weak ivariace priciple if we assume more for the trasformatio S. For each x [0, ) ad m N let us take the followig radom variables: W m,x ( l m ) = log µ 2(Q l (x)) l σ m ad we exted this liearly o each subitervals [ l m, l+ m ]. This way for each x W m,x (t) is a elemet of the space C[0, ] of the cotiuous fuctios o [0, ] topologised with the supremum orm. Defiitio 2.2. We say that S satisfies the weak ivariace priciple if the process W m (t) coverges i law to the Browia motio o [0, ]. I this sectio we are goig to suppose for the trasformatio S that it satisfies this weak ivariace priciple ad the, as we did it previously with the CLT-property, we are goig to prove that so does k(m, x). Our first theorem is the followig: Theorem 2.4. Suppose that the trasformatio T satisfies the 0-property ad that S satiesfies the weak ivariace priciple. Let us take K m,x ( l m ) = k(l, x) l σ m ad let us exted it liearly o each subitervals [ l m, l+ m ] i [0, ], where σ = σ. The the process K m(t) for t [0, ] coverges i law to the Browia motio o [0, ]. Proof. Sice S satiesfies the weak ivariace priciple, it is eough to show that sup K m (t) W m (t) 0 t [0,] i probability. Let us otice that both K m (t) ad W m (t) are costructed the same way that is we take their value at poits l m ad the exted them liearly i betwee. Hece it is sufficiet to take the supremum for t = l m for l = 0,,..., m. This is easy to see from the fact that the differece of two liear fuctios is liear ad that the supremum of a liear fuctio over a closed iterval is take at oe of the edpoits. So let us take 0 l m ad x [0, ). The we ca divide the differece ito three parts: ( ) ( ) l l K m,x W m,x = m m = k(l, x) l σ m log µ 2(Q l (x)) l σ m =

15 CHAPTER 2. EQUIPARTITIONS 2 = k(l, x) l + log µ 2(Q l (x)) + l σ m = = k(l, x) + log λ(p k(l,x)(x)) σ m λ(q l (x)) λ(p k(l,x) (x)) + log σ m µ 2(Q l (x)) λ(q l (x)) + log. σ m ( Let us deote this three parts with R l m, x) (, R l 2 m, x) ( ad R l 3 m, x) respectively. We are goig to prove that ( ) l l=0,,...,m R i m, x 0, i probability, for i=,2,3. The third term is the easiest: because of the equivalece of the measures there exists K ad K 2 that K µ2(b) λ(b) K 2 for every B Borel-set ad thus ( ) 0 l l=0,,...,m R 3 m, x { log K, log K 2 } σ m ad this goes to 0. For the other two terms we are goig to eed a little lemma: Lemma 2.2. Let g m (l) = m f(l) be a real valued fuctio for positive iteger m, where 0 l m is also a iteger. Let us suppose that g m (m) 0 as m. The 0 l m g m(l) 0 as m. Proof. First we are goig to prove that g m(l) 0 0 l m as m. Let l(m) be the largest iteger ot bigger tha m, where the imum occurs, i.e. g m (l(m)) = 0 l m g m(l). Let us otice the followig: g m+ (l(m + )) = { } m g m (l(m)), g m+ (m + ). m + From this it is easy to see that either l(m + ) = l(m) or l(m + ) = m +. Hece wheever l(m + ) > l(m) the l(m + ) = m +. Thus l(m) is mootoe icreasig ad if it is ot bouded, the there are ifiitely may m itegers for which l(m) = m. This way we ca distiguish two cases: the first is whe l(m) is bouded: this ca oly happe if there exists N iteger that for every m N: l(m) = l(n). So let us take i this case m N: g m (m) 0 l m g m(l) = g m (l(m)) = g m (l(n)) = N m g N (l(n)).

16 CHAPTER 2. EQUIPARTITIONS 3 Here both the first ad the last term goes to 0 as m ad so does the imum. The secod case is whe l(m) is ot bouded: this ca oly happe if there is a strictly icreasig (ad thus goig to ifiity) sequece m such that l(m ) = m ad if m m < m +, the l(m) = l(m ) = m. This way: g m (m) g m m(l) = g m (l(m)) = g m (m ) = g m (m ) g m (m ) 0 l m m here agai the first ad the last term goes to 0 as m. Now let us otice the followig: g m(l) = { g m(l), mi g m(l)}. 0 l m 0 l m 0 l m We ca use the first part of the proof for the fuctio r = g ad coclude that which fiishes the proof. r m(l) = mi g m(l) 0, 0 l m 0 l m We ca use this lemma to coclude the proof of our theorem. First we show that ( ) l l=0,,...,m R m, x 0 almost everywhere. Let us take x [0, ) where the covergece of the 0- property ad the covergece of Theorem 2.2 hold. Both happe almost everywhere, so it is sufficiet to prove the above covergece for these x. Now we ca take f(l) = k(l, x) + log λ(p k(l,x)(x)) σ. We ca use the lemma with this f because we assumed that at the poit x the 0-property holds, i.e.: log µ (P (x)) 0. As usual, because of the equivalece of the measures λ ad µ the same holds if we write λ istead of µ i the covergece of the 0-property. We eed to check if the assumptio of the lemma holds: g m (m) = m f(m) = k(m, x) + log λ(p k(m,x)(x)) σ m = = k(m, x) m which goes to 0 as m. k(m, x) + log λ(p k(m,x) (x)) σ k(m, x),

17 CHAPTER 2. EQUIPARTITIONS 4 Next we show that l=0,,...,m R 2 ( ) l m, x 0 i probability. Here we take x [0, ) - bearig i mid Remark such that the covergece i Lemma 2. holds. The we take f(l) = λ(q l (x)) log σ λ(p k(l,x) (x)). Now we ca agai use the previous lemma if we show that its assumptio holds: g m (m) = m f(m) = σ m log λ(q m (x)) λ(p k(m,x) (x)), which goes to 0 because of Lemma 2.. At last we ca put all the pieces together: ( ) ( 0 sup K m,x (t) W m,x (t) = l l t [0,] l=0,,...,m K m,x W m,x m m) l=0,,...,m R ( ) l m, x + l=0,,...,m R 2 i probability. This proves the theorem. ( ) l m, x + l=0,,...,m R 3 ( ) l m, x 0 We are goig to use the previous theorem to get a similar result for m(, x). To get there, first we eed a few propositios as we are goig step by step. Propositio 2.. Let us suppose that we have a stochastic process S t for t 0, with cotiuous realizatios. Let us take S (t) = S t. If S (t) coverges i law to the Browia motio o [0, ], the it also coverges i law to it o [0, k] for every k N. Proof. Let us take φ: C([0, ]) C([0, k]) such that φ(ω(t)) = kω ( ) t ) k. The this φ is a cotiuous mappig. Let us otice that φ (S k [0,] = S [0,k]. It is well-kow that if B(t) is a Browia motio, the so is ab ( ) t ) a for every 2 a 0. This meas that φ (B [0,] = B [0,k] i law, ad thus it is a Browia motio o [0, k]. Hece as φ is cotiuous ad S k [0,] coverges i law to the Browia motio o [0, ], S [0,k] coverges i law to the Browia motio o [0, k]. This cocludes the proof. This propositio tells us that if we exted the defiitio of K, the it follows that it coverges i law to the Browia motio o [0, k]. This observatio will

18 CHAPTER 2. EQUIPARTITIONS 5 be of great help i the sequel. But first of all, let us itroduce the followig two processes: K,x (t l (x)) = l k(l, x) σ ad M,x (t l (x)) = m(k(l, x), x) k(l, x) σ for t k(l,x) l (x) = ad for l = 0,,..., (x) +, where (x) is the largest positive iteger for which t (x) (x). The as before we exted these liearly o each subitervals [t l (x), t l+ (x)] for l = 0,,..., (x) ad we take the parts for t [0, ]. Propositio 2.2. Suppose that the trasformatio T satisfies the 0-property ad that S satiesfies the weak ivariace priciple. The the process K (t) coverge i law to the Browia motio o [0, ]. Proof. We will show that K (t) has the same limit as K (t). As both of them are piecewise liear, the imum of the differece ca oly occur at poits l for l {0,..., } or at t l (x) for l {0,..., (x)}. Let us also otice that because of the defiitios of the processes ( ) l K,x (t l (x)) = K,x. Thus we ca write the differece betwee K ad K at poits t l (x) for l = 0,..., (x) as the differece of the value of K at two certai poits: ( ) l K,x (t l (x)) + K,x (t l (x)) = K,x (t l (x)) K,x. We would like to do the same for the poits l too. Let us itroduce l(, x) = {l : t (x) > l (x) }. For every l l(, x) there exists a l < (x) such that t l (x) l < t l + (x). For l(, x) < l we ca take l = (x) as t (x) l (x) < t (x). From the costructio of the process (x)+ K it follows that ( ) l K,x(t l (x)) K,x K,x(t l +(x)) or ( ) l K,x(t l (x)) K,x K,x(t l +(x)). The as K,x(t l (x)) = K,x ( l ) we get that ( ) l K,x K,x ( ) ( l l ) + K,x

19 CHAPTER 2. EQUIPARTITIONS 6 or ( ) l K,x K,x sup t [0,] ( ) ( l l ) + K,x. Now we ca put the above pieces together to get K,x (t) + K,x (t) { ( ) ( ) } l l K,x + K,x + l=0,..., where ad l=0,..., (x) K,x (t l (x)) + K,x (t l (x)) l=0,..., {R (l, l, x)} + l=0,..., {R 2 (l, l, x)} + l=0,..., {R 3 (l, x)}, (x) ( ) ( ) R (l, l, x) = l l K,x K,x, ( ) ( ) R2 (l, l, x) = l l K +,x K,x, ( R3 (l, x) = l K,x (t l (x)) K,x. ) From the previous theorem we kow that K (t) coverges i law. From the tightess of the sequece it follows that for every η > 0 there exists a compact set K η C[0, ] that µ 2 (x [0, ]: K,x (t) K η ) > η, for each. From the compactess of a set K it follows that for ε > 0 exists δ > 0 such that w K implies that w(t) w(s) ε if t s < δ. Hece i order to prove that the sum above goes to 0 i probability, it is eough to show that { l l } ad l=0,..., 0 { t l (x) l } l=0,..., (x) 0 i probability i probability. First we are goig to eed a little lemma. Sice we will use it later i a more geeral form that is how we state it. Let us itroduce - aalogously to (x) - the otatio k (x) = {l 0: t l (x) k}. Lemma 2.3. lim λ(x [0, ]: k(x) < 2k) =.

20 CHAPTER 2. EQUIPARTITIONS 7 Proof. We eed to show that if we take ε > 0, N such that for every N: From Theorem 2. we kow that λ(x [0, ]: k (x) < 2k) > ε. lim k(, x) = a.e. The from Jegorov s theorem it follows that the covergece is also true almost uiformly, meaig that for every ε > 0 there exists A k [0, ] such that λ(a k ) < ε ad for η > 0 M such that for m M ad x [0, ] \ A k : Let us take η = 4k k(m, x) m > η. ad m = 2k. The we have k(2k, x) > 2k 2 > k. As t k (x) (x) k, we get for N = M 2k ad x [0, ] \ Ak k (x) < 2k. With this lemma, it is sufficiet to show that o the set [0, ] \ A { t l (x) l } { l=0,..., (x) t l (x) l } l=0,...,2 0 i probability. This follows easily from Lemma 2.2. We ca take the fuctio f as k(l, x) l f(l) = ad we shall remark that the lemma is also true ad ca be easily prove if we multiply f with m istead of m. Thus we have that if f(2) { t l (x) l } l=0,..., This is easy to see: f(2) = k(2, x) 2 = = k(2, x) 2 0 a.e.

21 CHAPTER 2. EQUIPARTITIONS 8 from Theorem 2.. Now from the defiitio of l we have that k(l,x) l < k(l +,x). Hece we ca write: l l l k(l, x) + k(l, x) l k(l +, x) k(l, x) k(l, x) l +. Thus for x [0, ] \ A { l l } l=0,..., k(l +, x) k(l, x) k(l, x) l + l =0,..., (x) l =0,...,2 k(l +, x) k(l, x) k(l, x) l +. Here agai we ca use the modificatio of Lemma meaig that we divide by ad ot by - with f (l) = k(l +, x) k(l, x) ad f 2 (l) = k(l, x) l. We oly have to check that f (2) = k(2 +, x) k(2, x) ad f 2 (2) k(2, x) 2 = 0 Both of them follow easily from Theorem 2.. With this we have just fiished provig that K,x (t) + K,x (t) 0 sup t [0,] 0 which meas that K also coverges weakly to the Browia motio o [0, ]. a.e. a.e. a.e. To take the ext step we eed a propositio which will help to uderstad the strog coectio betwee k(m, x) ad m(, x).

22 CHAPTER 2. EQUIPARTITIONS 9 Propositio 2.3. Assume that T satisfies the 0-property ad that S satisfies the weak ivariace priciple. The, 0 m(k(, x), x) o ( ), i probability. Proof. It is clear from the defitios that m (, x) = m(k(, x), x) for all N. From Lemma 2. it is also clear that ( ) λ(q (x)) log λ(q (x)) log λ(q m (,x)(x)) log = o ( ) λ(p k(,x) (x)) i probability. Sice µ 2 is equivalet to λ, we have that Thus lim log µ 2(Q (x)) log λ(q (x)) = log dµ 2 dλ (x). log µ 2 (Q (x)) log µ 2 (Q m lim (,x)(x)) = 0 i measure (both λ ad µ 2 ). Let us take W (t) as i Defiitio 2.2. From the weak ivariace priciple (2.3) W W, as where W is a Browia motio o [0, ]. As m (,x), for large eough m (, x) < 2 o a large set of x-es, i.e. for ε > 0 exists 0 such that µ 2 (x: m (, x) < 2) > ε if > 0. From the tightess of the sequece (W ), for ay η > 0 there is a compact set K η C([0, ]) such that µ 2 (x: W (., x) K η ) > η, for each. (2.4) From the compactess of K η it follows that for ε > 0 exists δ > 0 such that w K η implies that w(t) w(s) ε if t s < δ. Thus, W ( ) 2, x ( m (, x) W, x) ε 2 provided that so large that m (, x) < ( + 2δ), ad W (., x) K η. As m (, x) = 2 = log µ 2(Q (x)) + log µ 2 (Q m (,x)(x)) + m (, x) 2 +

23 CHAPTER 2. EQUIPARTITIONS 20 ( ) 2 = (W 2, x + log µ 2(Q (x)) log µ 2 (Q m (,x)(x)) 2 = ( m )) (, x) W 2, x + log µ 2(Q (x)) log µ 2 (Q m (,x)(x)), 2 2 ad here both terms coverges to 0 i probability, which we ca see from (2.3) ad (2.4). Hece so does m (,x). This completes the proof. With this the ext step becomes quite easy: Propositio 2.4. Suppose that the trasformatio T satisfies the 0-property ad that S satiesfies the weak ivariace priciple. The the process M (t) coverges i law to the Browia motio o [0, ]. Proof. We see that differece betwee K ad M, ad we otice that the supremum of the differece ca oly occur at poits t l (x). So for x [0, ] \ A sup M,x(t) K,x(t) m(k(l, x), x) l t [0,] l=0,..., (x) σ l=0,...,2 σ m(k(l, x), x) l. We ca agai use Lemma 2.2 by puttig f(l) = m(k(l, x), x) l ad the from Propositio 2.3 we have that M,x(t) also coverges to the Browia motio o [0, ], sice m(k(2, x), x) i probability. Now we ca state the last theorem, which claims that uder the same assumptios as before m (x) also satisfies the weak ivariace priciple: Theorem 2.5. Suppose that the trasformatio T satisfies the 0-property ad that S satiesfies the weak ivariace priciple. Let us take k N such that < k, the let us take ( ) l M,x = m(l, x) l σ ad let us exted it liearly o each subitervals [ l, l+ ]. The the process M (t) coverge i law to the Browia motio o [0, k].

24 CHAPTER 2. EQUIPARTITIONS 2 Proof. We just eed to show that the differece betwee M,x (t) ad M,x(t) goes to 0 i probability. Let us itroduce the otatio L = k. We otice i this case that the supremum of the differece ca oly occur at poits l for l = 0,..., L or at poit k. We take k (x) = { l 0: t l (x) k < t l+ (x)}. Deote l k (, x) = {l L: t l k (x) (x) > }. For every l l k(, x) there exists a l < k (x) such that k(l, x) l < k(l +, x). For L l > l k (, x) there is k (x) such that k( k (x), x) l < k( k (x) +, x), i.e. for these poits we ca take l = k (x). So we get that Hece we have m(k(l, x), x) k(l +, x) σ m(l, x) l σ m(k(l +, x), x) k(l, x) σ. M,x(t l (x)) k(l +, x) k(l, x) σ M,x M,x(t l +(x)) + k(l +, x) k(l, x) σ. First let us show that the correctig terms above go to 0: { k(l +, x) k(l }, x) l=0,...,l ( ) l { } k(l +, x) k(l, x) 0 i probability. l=0,..., k (x) Because of Lemma 2.3, it is eough to show that l=0,...,2k { k(l +, x) k(l, x) } 0 i probability. Agai we ca use Lemma 2.2 ad we bear i mid Remark 2.2 ad thus we oly eed to show that k(2k, x) k(2k, x) 0 i probability. ( Notice that this term equals to K,x (2k) K 2k ),x + σ. This coverges to 0 i probability, which follows from the fact that K coverges weakly to the Browia motio. Hece we got that l=0,...,l M,x ( l ) ( ) l M,x

25 CHAPTER 2. EQUIPARTITIONS 22 where l=0,...,l {R (l, l, x), R2 (l, l, x)}, ( R (l, l, x) = M,x l ( R2 (l, l, x) = M,x l ) M,x (t l (x)) ) M,x ( t l +(x) ). As before, because of the tightess of the sequece M,x, we oly eed to show that { l k(l }, x) 0 i probability l=0,...,l ad that l=0,...,l { Both follow if we show that l=0,..., k (x) l=0,...,l { k(l } +, x) l 0 { i probability. k(l +, x) k(l }, x) } 0 i probability. k(l +, x) k(l, x) Agai it is eough to prove this for x [0, ] \ A k. Ad for these x s we ca use Lemma 2.2. Ad as before it is sufficiet to show that k(2k, x) k(2k, x) 0 which follows from Theorem 2. sice both terms coverge to 2k. We oly have to check the differece at poit k ow. Because of the defiitio of L, L+ a.e. > k, so the differece at poit k caot be bigger tha the imum L L+. We have already covered the, so let us take a look at the other oe. If there exists l such that of the differeces at poits poit L ad k(l, x) = L+, the the differece is 0. Let us also otice that k( k (x)+, x) L + > L k( k (x), x), thus if such a l exists, the l = k (x) + else k( k (x) +, x) > L +. I this case m(k( k(x), x), x) k( k (x) +, x) σ Hece we have m(l +, x) L + σ m(k( k(x) +, x), x) k( k (x), x) σ. M,x(t k (x) (x)) k( k(x) +, x) k( k (x), x) σ M,x ( ) L +

26 CHAPTER 2. EQUIPARTITIONS 23 M,x(t k (x)+ (x)) + k( k(x) +, x) k( k (x), x) σ. First agai let us see that the correctig terms above go to 0. From Lemma 2.3 it follows that for x [0, ] \ A k ad for N λ(x [0, ]: k (x) < 2k) > ε. The agai it is eough to show that the covergece i probability holds o this set. Thus { } k( k (x) +, x) k( k (x), x) k(l +, x) k(l, x) 0 l=0,...,2k i probability, which we have already show. Hece, as before, because of the tightess of the sequece M (t), we oly have to prove that {k t k (x) (x), t k (x)+ (x) k} t k (x)+ (x) t k (x) (x) 0 i probability. This is also easy to see: t k (x)+ (x) t k (x) (x) l=0,...,2k { } k(l +, x) k(l, x) 0 i probability, which we have also already see. Hece the differece at poit k also coverges to 0 i probability. This fiishes the proof. Fially, we preset a corollary, which states the cetral limit theorem for m(, x), ad thus i a way it is a geeralizatio of Faivre s theorem, see ([4]). Corollary 2.. Uder the assumptios of Theorem 2.5 m(, x) σ 2 N (0, ), ( ) 3σ where σ 2 = = σ ad is the covergece i law with 3 respect to the probability measure µ. Proof. From Theorem 2.5, we have that Hece if we divide by ( ) M,x = ( m(, x) N 0, ). σ, we get that m(, x) σ = m(, x) σ 2 N (0, ).

27 Chapter 3 Examples 3. CLT-property We start the sectio with a theorem which gives the CLT-property for a wide class of trasformatios (see []): Theorem 3.. Let T be a strogly mixig Markov shift, ad put σ 2 = lim V ar[ log µ(p (x))], where Var stads for the variace. If σ 2 0, the ({ lim µ x: log µ(p }) (x)) σ u = u 2π e x2 2 dx. Now let us take Example.4 with k = 2 ad p 0 = p, p = p ad suppose that p 2. We have see that this is a Markov shift ad we have prove that it is strogly mixig. We take the partitio with atoms A 0 = {y : y 0 = 0} ad A = {y : y 0 = }. The oly thig we have to check is that V ar[ log µ(p lim (x))] 0. So let us check it. Take x X, the P (x) = {y : y 0 = x 0,..., y = x }, sice P = {A: A = {y : y 0 = a 0,..., y = a } for (a 0,..., a ) {0, } }. This meas that µ(p (x)) = p i= xi ( p) i= xi. The by itroducig the radom variable Y with Y = i= x i, we have log µ(p (x)) = log py log ( p)( Y ), 24

28 CHAPTER 3. EXAMPLES 25 where Y is biomial with parameters ad p. Hece we ca easily compute V ar[ log µ(p (x))] : V ar[ log µ(p (x))] = = log 2 pv ar[y ] + log 2 ( p)v ar[ Y ] + 2 log p log ( p)cov(y, Y ). Here we kow that V ar[y ] = V ar[ Y ] = p( p), ad we ca easily compute Cov(Y, Y ) from the defiitio: Thus we have Cov(Y, Y ) = E(Y ( Y )) E(Y )E( Y ) = = E(Y ) E(Y 2 ) E(Y ) + E 2 (Y ) = V ar[y ] = p( p). V ar[ log µ(p (x))] Hece lim V ar[ log µ(p (x))] ( p = p( p)(log ( p) log p) 2 = p( p) log 2 p ( ) = p( p) log 2 p p 0 if p 2. Therefore the coditios of Theorem 3. are satisfied, hece the trasformatio has the CLT-property property Our first example is Example.2. For these trasformatios eve more is true. Take P = {[ k r, ) } k+ r : k = 0,,..., r, the log λ(p (x)) = for every ad for every x [0, ). To see this, first take x [0, ) which is ot i the form of k r for some k {0,,..., r }. The T geerates the digits of x i the umber system with base r: rt k x gives the k-th digit. This meas that for every rt k x rt k x rt x + r k x < r k + r. Sice [ rt k x, ) rt k x r k + rt x + r k r P, we have that [ rt k x P (x) = r k, rt k x rt x ) + r k + r. Hece log λ(p (x)) = log r = log r for almost every x [0, ), which meas from the Shao-McMilla-Breima theorem that = log r. Thus T satisfies the 0-property. Some of the -trasformatio as i Example.3 also satisfy the property. We take {[ k P =, k + ) } {[ )} : k = 0,,...,,. ).

29 CHAPTER 3. EXAMPLES 26 The big differece betwee the two types of examples i this sectio is that i this case the last iterval is what is called o-full, meaig that its measure is which is smaller tha that of the others which is. Nevertheless, we will prove the 0-property for some -trasformatio which have somethig similar to our first example. But first of all, let us look ito the partitios geerated by the -trasformatio, i geeral. We will see that what plays a very importat role i the study of the legth of the elemets of the partitios is the error of the expasios of. Let us defie T with T = [0, ), so the we ca talk about T i for i 0. This way we ca itroduce the followig otatios: T k a = for 0. The propositio below explais why these are so importat: Propositio 3.. Let us deote the set of possible leghts of -cyliders: A = A P λ(a). The { } ai A = : i = 0,,..., i. Proof. { We will } prove { the} statemet by iductio. First for = we have A =, a = 0, a. Now let us suppose that the statemet is true for some, ad let us ) try to prove for[ +. ) First take a arbitrary A P ad look at T [0, A. Sice for x 0, T x is simply x, this meas that ( )) λ T [0, A = λ(a), by usig the fact the every A P is a iterval { } (which is easy to see by iductio). Thus A a = : a A A +. We get [ ) the same result if we look at T A k for every k {0,,..., }., k+ [ The oly ( + )-cyliders that we still have to check are T A )., Let [ ) + a, + b. I [ ) A = [a, b). The if b, the T A, = ( [ )) this case λ T A, [ ) A. If a, the T A, =. [ ) The iterestig case is whe a < < b. The T A, = [ + a )., So this leads us to the questio: What poits ca be edpoits of a -cylider? Lemma 3.. The edpoits of -cyliders are elemets of the followig set, deoted by E : { } i k E =, k : i k {0,,..., } [0, ]. k

30 CHAPTER 3. EXAMPLES 27 { } Proof. E = 0,,...,,. The it is eough to show that T (E \) E +. Let the e = i k < for some i k k {0,,..., } for k =,...,. [ ) I this case, take j {0,,..., }: T e j, j+ = j + e. (It is easy to see that j + e is a iverse image of e ad it is i the iterval j [ ) [.) Now if e, the T e [, ) { +, j+, ) = else T e { = + e. I every case T e = j + e } [0, : j {0,,..., } ) i k k : i k {0,,..., } } [0, ) E +. From the proof it follows that ay e < edpoit of a -cylider i the form e = i k for some i k k {0,,..., } is i the iverse image T j ( i k k=j+ ) for ay j 0. k j Now we ca fiish the proof of Propositio 3.. The oly thig left that we eed to check is whe we have a cylider i the form [a, b) where a < < b. It meas that a is the biggest of the possible edpoits of -cyliders which is smaller tha. We kow from the previous lemma that a ca be writte i the form i k + a the form + for some i k k {0,,..., }. Now we also kow that is the biggest umber that is smaller tha amogst those who are i i k k for some i k {0,,..., }. Let us write the + a + = j k k. Itroduce the otatio b k = T k for the momet. We are goig to prove by iductio that j k = b k for k =, 2,..., for every. First let =. The of course j = = T 0 = b. Now suppose that we have that j k = b k for every k =, 2,..., j ad for every j, ad let us try to prove it for +. We are goig to prove that by cotradictio. Suppose that there are some i, i 2,..., i + such that ot all i k = b k ad that + + b k k < i k k <. Let j the smallest iteger for which i j b j ad suppose that j. The it is also true that + k=j + b k k j+ < k=j i k k j+. It follows from the iductive assumptios that b j > i j. If the opposite were true, the if we look at the case whe = j, the we would have a bigger choice the j b k, amely j i k k <. That is a cotradictio. So we have that k b j i j +. Thus we have that

31 CHAPTER 3. EXAMPLES 28 b j i j < + k=j+ i k b k k j + k=j+ i k k j. From the proof of Lemma 3. ad the remark after that, if + edpoit of a cylider, the it is a iverse image of T j ( + k=j+ i k k is a i k k j ) for ay j 0. But this cotradicts the fact that + i k k=j+ >. k j The oly possibility left is that j = +. The we caot use the iductive assumptios but we ca use the followig lemma, which will be useful later o too: Lemma 3.2. For every x [0, ] x = T k k x + T x. Proof. Agai by iductio. For = it follows from the defiitio of the trasformatio. Now suppose that it is true for some. Take +. The T + x = T x T x. Thus T T x = x + + T + x +. This lemma shows us that it caot be that b + < i + because the if we use the lemma for x = we get that = + b k k + T + + < b k k + b i k + = k <, which is agai a cotradictio. This proves that j k = b k for every k. Ad we eeded all this to get that + a + = b k k ad from this we have that the measure of the (+)-cylider is: ( + a ) = + b k = a k +, ad this proves Propositio 3.. From the precedigs, we ca prove the 0-property for the followig class of -trasformatio: those for which the digits of the expasio of is periodic, meaig that T j = T j+ for every j k ad with, i.e. = k i= T i i + k+ i=k T i i ( l= ) l.

32 CHAPTER 3. EXAMPLES 29 This follows from the fact that for every x [0, ]: x = T k x. k It is apparet from Propositio 3. that the importat thig to look at is a m. Let m k. The from the defiitio of a m ad from the special form of the expasio of : m a m = i= T i i = i=m+ T i i = i=m++ T i This meas that the possible log P m (x) mh(t ) values ca be: log i = a +m. a i m i m log = log a i + (m i) log m log = log a i i log, for i {0,,..., m}. But this gives oly at most k + differet values: for i = 0,,..., k +, as for i k + : log a i i log = log a i i log = log a i (i ) log. So let us take M = log a i i log = log a i i log. i {0,,...,k+ } i N Now let ε > 0. We ca take N = M 2 ε 2 M m ε. From this it follows that for every x [0, ) : hece the 0-property is satisfied. log λ(p m (x)) mh(t ) ε, m. The for every m N we have Just to have some specific examples: we ca take the golde mea, i.e. 2 = +. The = +, so we ca already suspect that this is goig to 2 satisfy the coditio. I fact, T 2 = ( ) mod, ad sice = =, so ( ) =, hece T 2 = 0 ad as the 0 is a fixpoit for the trasformatio, this has the property i questio.

33 Bibliography [] G. H. Choe: Computatioal Ergodic Theory, page 253, Spriger (2004). [2] K. Dajai: Ergodic Theory, Lecture otes, [3] K. Dajai, A. Fieldsteel: Equipartitio of iterval partitios ad a applicatio to umber theory, Proc. Amer. Math. Soc. 29 (200), o. 2, (electroic). [4] C. Faivre: A cetral limit theorem related to decimal ad cotiued fractio expasio, Arch. Math. (Basel) 70 (998), o. 6,