Dalibor Volný

A CENTRAL LIMIT THEOREM FOR FIELDS OF MARTINGALE DIFFERENCES

We prove a central limit theorem for stationary random fields of martingale differences f • T i , i ∈ Z d , where T i is a Z d action and the martingale is given by a commuting filtration. The result has been known for Bernoulli random fields; here only ergodicity of one of commuting transformations generating the Z d action is supposed.

Introduction

In study of the central limit theorem for dependent random variables, the case of martingale difference sequences has played an important role, cf. Hall and Heyde, [HaHe]. Limit theorems for random fields of martingale differences were studied for example by Basu and Dorea [BD], Morkvenas [M], Nahapetian [N], Poghosyan and Roelly [PR], Wang and Woodroofe [WaW]. Limit theorems for martingale differences enable a research of much more complicated processes and random fields. The method of martingale approximations, often called Gordin's method, originated by Gordin's 1969 paper [G1]. The approximation is possible for random fields as well, for most recent results cf. e.g. [WaW] and [VWa]. Remark that another approach was introduced by Dedecker in [D] (and is being used since); it applies both to sequences and to random fields.

For random fields, the martingale structure can be introduced in several different ways. Here we will deal with a stationary random field f • T i , i ∈ Z d , where f is a measurable function on a probability space (Ω, µ, A) and T i , i ∈ Z d , is a group of commuting probability preserving transformations of (Ω, µ, A) (a Z d action). By e i ∈ Z d we denote the vector (0, ..., 1, ..., 0) having 1 on the i-th place and 0 at all other places, 1 ≤ i ≤ d.

F i , i = (i 1 . . . , i d ) ∈ Z d , is an invariant commuting filtration (cf. D. Khosnevisan, [K]) if (i) F i = T -i F 0 for all i ∈ Z d , (ii) F i ⊂ F j for i ≤ j in the lexicographic order, and (iii) F i ∩ F j = F i∧j , i, j ∈ Z d , and i ∧ j = (min{i 1 , j 1 }, . . . , min{i d , j d }). If, moreover, E E(f |F i ) F j = E(f |F i∧j )
, for every integrable function f , we say that the filtration is completely commuting (cf. [G2], [VWa]). By F (q) l , 1 ≤ q ≤ d, l ∈ Z, we denote the σ-algebra generated by the union of all F i with i q ≤ l. For d = 2 we by F ∞,j = F

(2) j denote the σ-algebra generated by the union of all F i,j , i ∈ Z, and in the same way we define F i,∞ .

We sometimes denote f • T i by U i f ; f will always be from L 2 . We say that U i f , i ∈ Z d , is a field of martingale differences if f is F 0 -measurable and whenever i = (i 1 . . . , i d ) ∈ Z d is such that i q ≤ 0 for all 1 ≤ q ≤ d and at least one inequality is strict then

E(f | F i ) = 0. Notice that U i f is then F i -measurable, i = (i 1 . . . , i d ) ∈ Z d , and if j = (j 1 . . . , j d ) ∈ Z d is such that j k ≤ i k for all 1 ≤ k ≤ n and at least one inequality is strict, E(U i f | F j ) = 0.
Notice that by commutativity, if U i f are martingale differences then

E(f | F (q) -1 ) = 0 for all 1 ≤ q ≤ d. (f • T j e q ) j
is thus a sequence of martingale differences for the filtration of F (q) j . In particular, for d = 2, (f • T j e 2 ) is a sequence of martingale differences for the filtration of F ∞,j = F

(2) j . Recall that a measure preserving transformation T of (Ω, µ, A) is said to be ergodic if for any

A ∈ A such that T -1 A = A, µ(A) = 0 or µ(A) = 1. Similarly, a Z d action (T i ) i is ergodic if for any A ∈ A such that T -i A = A, µ(A) = 0 or µ(A) = 1.
A classical result by Billinsley and Ibragimov says that if (f • T i ) i is an ergodic sequence of martingale differences, the central limit theorem holds. The result does not hold for random fields, however.

Example. As noticed in paper by Wang, Woodroofe [WaW], for a 2-dimensional random field Z i,j = X i Y j where X i and Y j , i, j ∈ Z, are mutually independent N (0, 1) random variables, we get a convergence towards a non normal law. The random field of Z i,j can be represented by a non ergodic action of Z 2 : Let (Ω, µ, A) be a product of probability spaces (Ω ′ , µ ′ , A ′ ) and (Ω ′′ , µ ′′ , A ′′ ) equipped with ergodic measure preserving transformations T ′ and T ′′ . On Ω we then define a measure preserving Z 2 action T i,j (x, y) = (T ′ i x, T ′′ j y). The σalgebras A ′ , A ′′ are generated by N (0, 1) iid sequences of random variables (e ′ • T ′ i ) i and (e ′′ • T ′′ i ) i respectively. The dynamical systems (Ω ′ , µ ′ , A ′ , T ′ ) and (Ω ′′ , µ ′′ , A ′′ , T ′′ ) are then Bernoulli hence ergodic (cf. [CSF]). On the other hand, for any

A ′ ∈ A ′ , A ′ × Ω ′′ is T 0,1 -invariant hence T 0,1 is not an ergodic transformation.
Similarly we get that T 1,0 is not an ergodic transformation either. By ergodicity of T ′ , T ′′ , A ′ × Ω ′′ , A ′ ∈ A ′ , are the only T 0,1 -invariant measurable subsets of Ω and A ′′ × Ω ′ , A ′′ ∈ A ′′ , are the only T 1,0 -invariant measurable subsets of Ω (modulo measure µ). Therefore, the only measurable subsets of Ω which are invariant both for T 0,1 and for T 1,0 are of measure 0 or of measure 1, i.e. the Z 2 action T i,j is ergodic.

On Ω we define random variables X, Y by X(x, y) = e ′ (x) and Y (x, y) = e ′′ (y). The random field of (XY ) • T i,j then has the same distribution as the random field of Z i,j = X i Y j described above. The natural filtration of F i,j = σ{(XY ) • T i ′ ,j ′ : i ′ ≤ i, j ′ ≤ j} is commuting and ((XY ) • T i,j ) i,j is a field of martingale differences.

A very important particular case of a Z d action is the case when the σ-algebra A is generated by iid random variables

U i e, i ∈ Z d . The σ-algebras F j = σ{U i : i k ≤ j k , k = 1, . . . , d} are then a completely commuting filtration and if U i f , i ∈ Z d is a 2
martingale difference random field, the central limit theorem takes place (cf. [WW]). This fact enabled to prove a variety of limit theorems by martingale approximations (cf. e.g. [WaW], [VWa]).

For Bernoulli random fields, other methods of proving limit theorems have been used, cf. e.g. [ElM-V-Wu], [Wa], [BiDu].

The aim of this paper is to show that for a martingale difference random field, the CLT can hold under assumptions weaker than Bernoullicity.

Main result

Let T i , i ∈ Z d , be a Z d action of measure preserving transformations on (Ω, A, µ), (F i ) i , i ∈ Z d , be a commuting filtration. By e i ∈ Z d we denote the vector (0, ..., 1, ..., 0) having 1 on the i-th place and 0 at all other places, 1 ≤ i ≤ d.

Theorem. Let f ∈ L 2 , be such that (f • T i ) i is a field of martingale differences for a completely commuting filtration F i . If at least one of the transformations T e i , 1 ≤ i ≤ d, is ergodic then the central limit theorem holds, i.e. for n 1 , . . . ,

n d → ∞ the distributions of 1 √ n 1 . . . n d n 1 i 1 =1 • • • n d i d =1 f • T (i 1 ,...,i d )
weakly converge to N (0, σ 2 ) where σ 2 = f 2 2 . Remark 1. The results from [VoWa] remain valid for Z d actions satisfying the assumptions of the Theorem, Bernoullicity thus can be replaced by ergodicity of one of the transformations T e i . Under the assumptions of the Theorem we thus also get a weak invariance principle. [VoWa] implies many earlier results, cf. references therin and in [WaW].

Proof.

We prove the theorem for d = 2. Proof of the general case is similar. We suppose that the transformation T 0,1 is ergodic and f 2 = 1. To prove the central limit theorem for the random field it is sufficient to prove that for m

k , n k → ∞ as k → ∞, (1) 1 √ m k n k m k i=1 n k j=1 f • T i,j converge in distribution to N (0, 1).
Recall the central limit theorem by D.L. McLeish (cf. [M]) saying that if X n,i , i = 1, . . . , k n , is an array of martingale differences such that (i) max 1≤i≤k n |X n,i | → 0 in probability, (ii) there is an L < ∞ such that max 1≤i≤k n X 2 n,i ≤ L for all n, and (iii)

k n i=1 X 2 n,i → 1 in probability, then k n i=1 X n,i converge to N (0, 1) in law.
Next, we will suppose k n = n; we will denote U i,j f = f • T i,j . For a given positive integer v and positive integers u, n define

F i,v = 1 √ v v j=1 U i,j f, X n,i = 1 √ n F i,v , i = 1, . . . , n.
Clearly, X n,i are martingale differences for the filtration (F i,∞ ) i . We will verify the assumptions of McLeish's theorem. The conditions (i) and (ii) are well known to follow from stationarity. For reader's convenience we recall their proofs.

(i) For ǫ > 0 and any integer v ≥ 1,

µ( max 1≤i≤n |X n,i | > ǫ) ≤ n i=1 µ(|X n,i | > ǫ) = nµ 1 √ nv v j=1 U 0,j f > ǫ ≤ ≤ 1 ǫ 2 E 1 √ v M j=1 U 0,j f 2 1 | v j=1 U 0,j f |≥ǫ √ nv → 0
as n → ∞; this proves (i). Notice that that the convergence is uniform for all v.

To see (ii) we note

max 1≤i≤n |X n,i | 2 ≤ n i=1 X 2 n,i = 1 n n i=1 1 √ v v j=1 U i,j f 2 which implies E max 1≤i≤n |X n,i | 2 ≤ 1.
[WaW] It remains to prove (iii). Let us fix a positive integer m and for constants a 1 , . . . , a m consider the sums

m i=1 a i v j=1 U i,j f, v → ∞.
Then ( m i=1 a i U i,j f ) j , j = 1, 2, . . . , are martingale differences for the filtration (F ∞,j ) j and by the central limit theorem of Billingsley and Ibragimov [Bil], [I] (we can also prove using the McLeish's theorem)

1 √ v v j=1 m i=1 a i U i,j f converge in law to N (0, m i=1 a 2 i ).
Notice that that here we use the assumption of ergodicity of T 0,1 . From this it follows that the random vectors (F 1,v , . . . , F m,v ) where

F u,v = 1 √ v v j=1 U u,j f, u = 1, . . . , m,
converge in law to a vector (W 1 , . . . , W m ) of m mutually independent and N (0, 1) distributed random variables. For a given ǫ > 0, if m = m(ǫ) is sufficiently big then we have 1

-(1/m) m u=1 F 2 u,v 1 < ǫ/2.
Using a truncation anrgument we can from the convergence in law of (F u,v , . . . , F m,v ) towards (W 1 , . . . , W m ) deduce that for m = m(ǫ) sufficiently big and v bigger than some v(m, ǫ),

1 - 1 m m u=1 F 2 u,v 1 < ǫ.
Any positive integer N can be expressed as N = pm + q where 0 ≤ q ≤ m -1. Therefore

(2) 1 - 1 N N i=1 F 2 i,v = m N p-1 k=0 1 m (k+1)m i=km+1 F 2 i,v -1 + 1 N N i=mp+1 F 2 i,v - q N .
There exists an N ǫ such that for N ≥ N ǫ we have 1

N N i=mp+1 F 2 i,v -q N 1 < ǫ hence if v ≥ v(m, ǫ) and N ≥ N ǫ then (3) 1 - 1 N N i=1 F 2 i,v 1 = 1 - 1 N v N i=1 v j=1 U i,j f 2 1 < 2ǫ.
This proves that for ǫ > 0 there are positive integers v(m, ǫ/2) and N ǫ such that for M ≥ v(m, ǫ/2) and n ≥ N ǫ , for

X n,i = (1/ √ n)F i,M n i=1 X 2 n,i -1 1 = n i=1 1 √ nM M j=1 U i,j f 2 -1 1 < ǫ.
In the general case we can suppose that T e d is ergodic (we can permute the coordinates). Instead of T i,j we will consider transformations T i,j where i ∈ Z d-1 and in (3), instead of segments {km + 1, . . . , km + m} we take boxes of (k 1 m + i 1 , . . . , k d-1 m + i d-1 ), i 1 , . . . , i d-1 ∈ {1, . . . , m}.

This finishes the proof of the Theorem.

Remark 2. For any positive integer d there exists a random field of martingale differences (f • T i ) for a commuting filtration of F i where T i , i ∈ Z d , is a non Bernoulli Z d action and all T e i , 1 ≤ i ≤ d, are ergodic.

To show this we take a Bernoulli Z d action T i , i ∈ Z d on (Ω, A, µ) generated by iid random variables (e • T i ) as defined e.g. in [WaW] or [VWa]. Then we take another Z d action of irrational rotations on the unit circle (identified with the interval [0, 1)) generated by τ e i = τ θ i , τ θ i x = x + θ i mod 1; θ i , 1 ≤ i ≤ d, are linearly independent irrational numbers. The unit circle is equipped with the Borel σ-algebra B and the (probability) Lebesgue measure λ. On the product Ω × [0, 1) with the product σ-algebra and the product measure we define the product Z d action (T i × τ i )(x, y) = (T i x, τ i y). Because the product of ergodic transformations is ergodic, for every e i , 1 ≤ i ≤ d, T e i × τ e i is ergodic. The product Z d action is not Bernoulli (it has irrational rotations for factors).

On Ω × [0, 1) we define a filtration F (i 1 ,...,i d ) = σ{U (i ′ ,...,i ′ d ) e • π 1 , i ′ -1 ≤ i 1 , . . . , i ′ d ≤ i d , π -1 2 B} where π 1 , π 2 are the coordinate projection of Ω × [0, 1). The filtration defined above is commuting and we can find a random field of martingale differences satisfying the assumptions of the Theorem.

Remark 3. In the one dimensional central limit theorem, non ergodicity implies a convergence towards a mixture of normal laws. This comes from the fact that using a decomposition of the measure µ into ergodic components, we get the "ergodic case" for each of the components (cf. [V]); the variance is given by the limit of (1/n) n i=1 U i f 2 which by the Birkhoff Ergodic Theorem exists a.s. and in L 1 and is T -invariant. In the case of a Z 2 action (taking d = 2 for simplicity), the limit for T 0,1 need not be T 1,0 -invariant. This is exactly the case described in the Example and eventually we got there a convergence towards a law which is not normal.

Acknowledgement. I am very thankful to Jérôme Dedecker for his remarks, comments, and encouragement. I am also thankful to Zemer Kosloff; the idea/conjecture that it is the ergodicity of coordinate factors of the Z d action which can imply the central limit theorem came out first in our discussion after my lecture in April 2014.


Zemer Kosloff

Dalibor Volny

Dalibor Volny Local

research institutions in France or abroad, or from public or private research centers. LOCAL LIMIT THEOREM IN DETERMINISTIC SYSTEMS

We show that for every ergodic and aperiodic probability preserving system, there exists a Z valued, square integrable function f such that the partial sums process of the time series f • T i ∞ i=0 satisfies the lattice local limit theorem.

introduction

Given an ergodic, aperiodic and probability measure preserving dynamical system (X, B, m, T ), we show the existence of a measurable square integrable function f with zero mean such that its corresponding ergodic sums process S n (f ) := n-1 k=0 f • T k satisfies the lattice local central limit theorem. A centered function f : X → R satisfies the central limit theorem if for all u ∈ R m S n (f )

S n (f ) 2 ≤ u ----→ n→∞ u -∞ e -x 2 2 dx. A function f : X → Z with f dm = 0 such that lim n→∞ Sn(f ) 2 2 n = σ 2 > 0 satisfies a lattice local central limit theorem if sup x∈Z √ n • m (S n (f ) = x) - e -x 2 /(2nσ 2 ) √ 2πσ 2 ----→ n→∞ 0.
There is also a non-lattice version of the local limit theorem which we do not consider in this paper. A function which satisfies the central limit theorem will be called a CLT function and if f satisfies a local central limit theorem (whether lattice or non-lattice) we will say that f is a LCLT function.

In case the measure theoretic entropy of the system is positive, if follows from the Sinai factor theorem that there exists a function f : X → Z taking finitely many values such that the sequence {f • T n } ∞ n=0 is distributed as an i.i.d. sequence. From this it is easy to construct a LCLT function. The question of existence of a central limit function for a zero entropy system such as an irrational rotation is more subtle. In the case of certain zero entropy Gaussian dynamical systems, Maruyama showed in [START_REF] Maruyama | Nonlinear functionals of Gaussian stationary processes and their applications[END_REF] existence of CLT functions such that the variance of S n (f ) grows linearly with n. Some years later, the seminal paper of Burton and Denker [START_REF] Burton | On the central limit theorem for dynamical systems[END_REF] showed that for every aperiodic dynamical system there exists a function f which satisfies the central limit theorem. By [START_REF] Volný | On limit theorems and category for dynamical systems[END_REF], the set of CLT functions is a meagre set in L 2 0 := f ∈ L 2 (m) : f dm = 0 . See also [START_REF] Liardet | Sums of continuous and differentiable functions in dynamical systems[END_REF] and [START_REF] Durieu | On sums of indicator functions in dynamical systems[END_REF] for such results in other function spaces. Following [START_REF] Burton | On the central limit theorem for dynamical systems[END_REF], several extensions and improvements regarding existence and bounds of the regularity of CLT functions were done, see for example [START_REF] Kato | Central limit theorem for Weyl automorphism[END_REF], [START_REF] Lacey | On weak convergence in dynamical systems to self-similar processes with spectral representation[END_REF], [START_REF] Lacey | On central limit theorems, modulus of continuity and Diophantine type for irrational rotations[END_REF], [START_REF] De La Rue | On the central limit theorem for aperiodic dynamical systems and applications[END_REF]. The most relevant to this work is [START_REF] Volný | Invariance principles and Gaussian approximation for strictly stationary processes[END_REF] where for every aperiodic dynamical system, a function satisfying the invariance principle and the almost sure invariance principle was constructed. More recently the question of weak convergence to other distributions was studied in [START_REF] Aaronson | Distributional limits of positive, ergodic stationary processes and infinite ergodic transformations[END_REF], [START_REF] Thouvenot | Limit laws for ergodic processes[END_REF]. In the dynamical systems setting, it is in general a nontrivial problem to determine whether a function which satisfies the central limit theorem also satisfies the local central limit theorem. In fact, even in the nicer setting of chaotic (piecewise) smooth dynamical systems a local CLT is usually proved under more stringent spectral conditions, see for example [START_REF] Rousseau-Egele | Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux[END_REF], [START_REF] Guivarc | Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov[END_REF], [START_REF] Aaronson | Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps[END_REF], [START_REF] Aaronson | Aperiodicity of cocycles and conditional local limit theorems[END_REF] and [START_REF] Gouëzel | Berry-Esseen theorem and local limit theorem for non uniformly expanding maps[END_REF]. The methods of proving the central limit theorem in [START_REF] Burton | On the central limit theorem for dynamical systems[END_REF] and [START_REF] Volný | Invariance principles and Gaussian approximation for strictly stationary processes[END_REF] involved non-spectral tools which are not adapted for getting a local CLT. Moreover, the resulting partial sum process of the function takes uncountably many values. Our main theorem is the following.

Theorem (See Theorem 4). For every ergodic, aperiodic and probability measure preserving dynamical system (X, B, m, T ) there exists a square integrable function f : X → Z with f dm = 0 which satisfies the lattice local central limit theorem.

The construction of the aforementioned function relies on a new version of the stochastic coding theorem [START_REF] Grillenberger | On marginal distributions and isomorphisms of stationary processes[END_REF], [START_REF] Kieffer | On coding a stationary process to achieve a given marginal distribution[END_REF]. Namely we show that in any ergodic, aperiodic dynamical system we can realize every independent triangular array which takes finitely many values, see Proposition 1. We remark that for the construction of the function f we need to realize a variant of a triangular array, see the beginning of Section 3 for the details.

Notation. For z ≥ 0, the expression x = y ± z stands for |x -y| ≤ z. Similarly x = ae ±b means ae -b ≤ x ≤ xe b . For sequences a(n), b(n) > 0 we will write a(n) ∼ b(n) if lim n→∞ a(n) b(n) = 1.
In some cases it will be denoted with the little o notation, that is a

(n) = b(n) + o(1). By a(n) = O (b(n)) and a(n) b(n) we mean lim sup n→∞ a(n) b(n) < ∞.
For convenience in the arithmetic arguments, log(•) denotes logarithm to the base 2 and ln(•) is the standard logarithm.

The strong Alpern tower lemma and realizations of triangular arrays

Let A be a finite set, {d n } ∞ n=1 an integer valued sequence and X n,m , n ∈ N, 1 ≤ m ≤ d n be an A-valued triangular array. In our setting this means

• For every n ∈ N and m ∈ {0, ..., d n -1}, X n,m : (Ω, F, P) → A.

• For every n ∈ N, X n,1 , . . . , X n,dn are identically distributed.

• {X n,m } n∈N,m∈{1,..,dn} is an independent array of random variables.

This section is concerned with the following realization of triangular arrays in arbitrary ergodic measure preserving transformations. See [START_REF] Grillenberger | On marginal distributions and isomorphisms of stationary processes[END_REF], [START_REF] Kieffer | On coding a stationary process to achieve a given marginal distribution[END_REF] for results in a similar flavor.

Proposition 1. Let (X, B, µ, T ) be an ergodic invertible probability measure preserving transformation. For every finite set A, and an A valued triangular array X := {X n,m } n∈N,m∈{0,..,dn} , there exists a sequence of functions f n : X → A such that {f n • T m } n∈N,m∈{0,..,dn-1} and X have the same distribution.

The construction of the functions f n is by induction on N ∈ N so that {f n • T m } 1≤n≤N,m∈{0,..,dn-1} and {X n,m } 1≤n≤N,m∈{1,..,dn} have the same distribution. For a fixed N one can consider ξ, the finite partition of X according to the value of the vector valued function

(G N ) n,m (x) := f n • T m (x), 1 ≤ n ≤ N, 0 ≤ m ≤ d n -1.
For this reason the previous Proposition is a corollary of the following proposition.

Proposition 2. Let (X, B, m, T ) be an ergodic invertible probability measure preserving transformation and ξ a finite measurable partition of X. For every finite set A and X 1 , X 2 , .., X l a collection of A valued i.i.d. random variables, there exists f :

X → A such that f • T i l-1 i=0
and {X i } l i=1 are equally distributed and f • T i l-1 i=0 is independent of ξ. The proof makes use of Alpern-Rokhlin castles. Given N ∈ N, a {N, N + 1} Alpern-Rokhlin castle for (X, B, m, T ) is given by two measurable sets B N , B N +1 such that • T j B L : L ∈ {N, N + 1}, 0 ≤ j < L is a partition of (X, B, m) to pairwise disjoint sets.. • We call B N , B N +1 the base elements of the castle and the atoms in the corresponding partition are referred to as rungs. We call T N B n T N +1 B N +1 the top of the castle.

Given ξ, a finite partition of X, a {N, N + 1} castle is ξ independent if every rung in the castle is independent of ξ. All equalities of sets mean equality modulo null sets.

Proof. By [START_REF] Campbell | Independence and Alpern multitowers[END_REF][Corollary 1], there exists a ξ-independent {2l, 2l + 1} castle with bases B = B 2l and F = B 2l+1 . Note that as T is invertible then,

T T 2l-1 B T 2l F = B F.
Let ζ 1 be the partition of the top of the tower {T 2l-1 B, T 2l F } which is its refinement according to ∨ 2l i=0 T i ξ. First we define f :

∪ T -j C : C ∈ ζ 1 , 0 ≤ j < l → A as follows. Partition each C ∈ ζ 1 to A l elements C a : a ∈ A l such that for every a = (a i ) l-1 i=0 ∈ A l , m (C a ) = m(C) l-1 i=0 P (X i+1 = a i ) . and set f (T -l+1+i x) = a i , i = 0, . . . , l -1, if there exists a ∈ A l and C ∈ ζ 1 such that x ∈ C a .
It remains to define f in the bottom rungs of the tower. By ζ 1 we denote the refinement of ζ 1 by the sets C a . By ζ we denote the joint partition of the base of the castle by

T -2l+1 (ζ 1 ∩ T 2l-1 B), T (ζ 1 ∩ T 2l-1 B), T -2l (ζ 1 ∩ T 2l F ), T (ζ 1 ∩ T 2l F ).
For every C ∈ ζ we do as follows. If C ⊂ B then we partition C to A l elements C a : a ∈ A l such that for every a = (a i )

l-1 i=0 ∈ A l , m (C a ) = m(C) l-1 i=0 P (X i+1 = a i ) and set f (T i x) = a i if there exists a ∈ A l and C ∈ ζ such that x ∈ C a .
If C ⊂ ζ ∩ F we do the same with l + 1 replacing l.

For any C ∈ ζ 1 we thus have

m (T -l+1 C) ∩ ∩ l-1 i=0 f • T i = a i+1 = m(T -l+1 C) l i=1 P (X i = a i ) , for C ∈ ζ ∩ B we have m C ∩ ∩ l-1 i=0 f • T i = a i+1 = m(C) l i=1 P (X i = a i ) ,
and for

C ∈ ζ ∩ F we have m C ∩ ∩ l i=0 f • T i = a i+1 = m(C) l+1 i=1 P (X i = a i ) . Moreover, for C ∈ ζ 1 ∩ T 2l-1 B the sets T -2l+1 C a are independent of (f • T i ) l-1 i=0 (conditionally on T -2l+1 C) hence m (T -2l+1 C) ∩ ∩ 2l-1 i=0 f • T i = a i+1 = m(C) 2l i=1 P (X i = a i ) (1)
and for

C ∈ ζ 1 with T -2l C ⊂ F we have m (T -2l C) ∩ ∩ 2l i=0 f • T i = a i+1 = m(C) 2l+1 i=1 P (X i = a i ) . ( 2
) Let C ∈ T -2l+1 (ζ 1 ∩ T 2l-1 B) or C ∈ T -2l (ζ 1 ∩ T 2l F ), 0 ≤ k ≤ l. We have T k C ∩ ∩ 2l-1-k i=0 f • T i = a i+k+1 = T k C ∩ ∩ 2l-1 i=k f • T i = a i+1 ;
by ( 1) and ( 2) we get

m (T k C) ∩ ∩ 2l-1-k i=0 f • T i = a i+k+1 = m(T k C) 2l i=k+1 P (X i = a i ) . ( 3
)
Let us show a similar equation for

T -l+1 C, C ∈ ζ 1 .
We have

T l (T -l+1 C a ) ∩ (T -l B)) ∩ ∩ 2l-1 i=l f • T i = a i+1 = (B ∩ T C a ) ∩ ∩ l-1 i=0 f • T i = a i+l+1
and the same equality holds for F , hence

m (T -l+1 C a ) ∩ (T -l B)) ∩ ∩ 2l-1 i=l f • T i = a i+1 = m (B ∩ T C a ) ∩ ∩ l-1 i=0 f • T i = a i+l+1 and m (T -l+1 C a ) ∩ (T -l F )) ∩ ∩ 2l-1 i=l f • T i = a i+1 = m (F ∩ T C a ) ∩ ∩ l-1 i=0 f • T i = a i+l+1 .
Therefore,

m (T -l+1 C a ) ∩ ∩ 2l-1 i=l f • T i = a i+1 = m (T C a ) ∩ ∩ l-1 i=0 f • T i = a i+l+1 and by independence of ζ and f, . . . , f • T l-1 on B ∪ F we deduce m (T -l+1 C a ) ∩ ∩ 2l-1 i=l f • T i = a i+1 = m(T C a )m (B ∪ F ) ∩ ∩ l-1 i=0 f • T i = a i+l+1 = m(C a ) l i=1 P (X i = a i+l+1 ) = m (C) 2l i=1 P (X i = a i ) . Recall that if C ∈ ζ 1 and a = (a i ) l i=1 ∈ A l then T -l+1 C a = T -l+1 C ∩ ∩ l-1 i=0 f • T i = a i+1 . Therefore, m (T -l+1 C) ∩ ∩ 2l-1 i=0 f • T i = a i+1 = m (T -l+1 C a ) ∩ ∩ 2l-1 i=l f • T i = a i+1 = m (C) 2l i=1 P (X i = a i ) , hence m (T -l+1 C) ∩ ∩ 2l-1 i=0 f • T i = a i+1 = m((T -l+1 C)) 2l i=1 P (X i = a i ) . For 0 ≤ k ≤ l -1 we thus have m (T -l+k+1 C) ∩ ∩ 2l-1 i=0 f • T i = a i+1 = m((T -l+1 C)) 2l i=1 P (X i = a i ) . ( 4
)
We show that f satisfies the conclusion of the proposition. Let D ∈ ξ and a ∈ A l . We claim that From ( 3) and ( 4) it follows that

m D ∩ ∩ l-1 i=0 f • T i = a i+1 = m(D) l i=1 P (X i = a i ) . ( 5
m C ∩ ∩ l-1 i=0 f • T i = a i+1 = m(C) l i=1 P (X i = a i ) . (6) Because ζ 1 is finer than (T 2l-1 B ∪ T 2l F ) ∩ ∨ 2l i=0 T i ξ, D ∩ Y is a union of sets C .
D is just a disjoint union of the sets C over all rungs Y . Summing equations ( 6) for all such C we see that equation ( 5) holds.

Definition of the function and proof of the CLT

Let (X, B, µ, T ) be an ergodic invertible probability measure preserving transformation and U : L 2 (X, µ) → L 2 (X, µ) is its corresponding Koopman operator. In this susbsection we construct the function f for which the local limit theorem holds. For k ∈ N we define

p k := 2 k , k even 2 k-1 + 1, k odd.
and

d k := 2 k 2
, and

α k := 1 p k √ k log k .
By a repeated inductive iteration of Proposition 2, there exists a sequence of functions fk :

X → {-1, 0, 1} such that: (a) For every k ∈ N, fk • T j 2d k +p k j=0
is an i.i.d. sequence, {-1, 0, 1} valued and

µ fk = 1 = µ fk = -1 = α 2 k 2 .
(b) For every k ≥ 2, the finite sequence fk • T j 2d k +p k j=0 is independent of

A k := fl • T j : 1 ≤ l < k, 0 ≤ j ≤ 2d k + p k .
To explain how we apply the proposition, the values of the functions g ∈ A k induce a partition ξ of X into 3 2d k +p k elements. Write X 1 , X 2 , X 3 , ..., X 2d k +p k for an i.i.d. sequence such that X 1 is {-1, 0, 1} distributed with E (X 1 ) = 0 and E X 2 1 = α 2 k and apply Proposition 2. In this definition the sequence of functions

fk • T j : k ∈ N, 0 ≤ j ≤ 2d k + p k
f k := p k -1 i=0 U i fk -U d k p k -1 i=0 U i fk .
It is worth to note that each of the function f k is a coboundary with transfer function

g k := d k -1 j=0 p k -1 i=0 U i+j fk , f k = g k -U g k . Proposition 3. f ∈ L 2 (X, µ). Proof. Fix k ∈ N and write V k,i := U i fk -U d k +i fk where i ∈ {0, 1, ..., p k -1}. This is a sequence of i.i.d. random variables with X V k,i dµ = 0 and V k,i 2 2 = 2α 2 k . Therefore f k 2 2 = X p k -1 i=0 V k,i 2 dm = p k -1 i=0 X (V k,i ) 2 dm = 2α 2 k p k ≤ 2 p k .
By condition (b) in the definition of the fk 's, the functions f 1 , f 2 , .. are independent. As they are also centered,

f 2 2 = ∞ k=1 f k 2 2 ≤ ∞ k=1 2 p k < ∞.
We conclude that f is well defined.

Theorem 4. The function f : X → R satisfies the local limit theorem with σ 2 := 2(ln 2) 2 . That is

sup x∈Z √ nµ (S n (f ) = x) - e -x 2 /(2nσ 2 ) √ 2πσ 2 ----→ n→∞ 0
Discussion on the steps in the proof of Theorem 4. The beginning of the proof is done by an argument similar to the one in Terrence Tao's blogpost on local limit theorems. The first step, which is done in the next subsection, is to prove the central limit theorem. This is done by calculating the second moments and verifying the Lindeberg condition. The choice of d k instead of an exponential sequence as in [START_REF] Volný | Invariance principles and Gaussian approximation for strictly stationary processes[END_REF] is used in this step.

In the course of the proof of the CLT, S n (f ) is decomposed into a sum of several independent random variables and we identify the main term, which we will call in this discussion Y n . By Proposition 16, the local limit theorem for S n (f ) is equivalent to the local limit theorem for the main term Y n .

In Section 4, we show the local limit theorem for Y n . There we use Fourier inversion and the CLT to reduce the local limit theorem to a question about uniform integrability of certain functions. In this step the choice of p k will help with a strong aperiodicity type statement which appears in the proof of Lemma 12. One problem we encounter, which is not present in the proof of classical local limit theorems, is that it seems a difficult problem to control the Fourier expansion of Y n around zero for an interval of fixed size (or a scaled interval of length constant times √ n). We overcome this problem by obtaining a sharp enough aperiodicity bound which reduces the estimate around zero to an interval of length constant times 4 √ n as in Lemma 13.

3.1. Proof of the CLT. We start by presenting S n (f ) as a sum of three terms depending on the scale of k with respect to n. That is

S n (f ) = Z Sm (n) + Ŷ (n) + Z La (n).
where

Z Sm (n) := k: d k ≤n S n (f k ) Ŷ (n) := k: p k <n<d k S n (f k ) Z La (n) := k: n≤p k S n (f k ) Lemma 5. For every n ∈ N the random variables Z Sm (n), Ŷ (n), Z La (n) are independent and (a) Z Sm (n) + Z La (n) 2 2 = O n √ log n . (b) 1 √ n Ŷ (n) -1 √ n S n (f ) ----→ n→∞ 0 in L 2 (X, µ). Proof. For each k ∈ N, S n (f k ) is a sum of functions from the sequence fk • T i n+d k +p k -1 i=0 .
For all k's appearing in the sums describing Ŷ (n) and Z La (n), one has n < d k , therefore Ŷ (n) and Z La (n) are sums of functions of the form fk

• T i 2d k +p k i=0 with k > √ log n and Z Sm (n) is a sum of functions from fk • T i 2d k * +p k * i=0
with k ≤ √ log n and k * is the smallest integer such that k * > √ log n. By property (b) in the definition of the fk 's we see that Z Sm (n) is independent of Ŷ (n) and Z La (n). A similar reasonning using property (b) of the functions fk shows that Z Sm (n), Ŷ (n), Z La (n) are independent. We now turn to prove part (a). Since

f k = g k -U g k then Z Sm (n) = k: d k ≤n (g k -U n g k ) . By independence of { fk • T i } p k +d k i=0 , g k 2 2 = d k +p k -1 j=0 (# {(l, i) ∈ [0, d k ) × [0, p k ) : l + i = j}) 2 fk 2 2 < α 2 k p 2 k (p k + d k ) ≤ 2 k 2 +1 k log k .
The functions {g k } ∞ k=1 are centered and independent, thus

k: d k ≤n g k 2 2 = k: d k ≤n g k 2 2 ≤ 2 k:d k ≤n 2 k 2 k log k = 2 k:k≤ √ log n 2 k 2 k log k = O n √ log n ,
where the last inequality follows from Lemma 17. As U is unitary it follows that

Z Sm (n) 2 2 = O n √ log n . ( 7
)
It remains to bound Z La (n) 2 2 . First note that for k such that n ≤ p k , by independence of the summands

S n (f k ) 2 2 = 2 p k -1+n j=0 fk 2 # {(i, l) ∈ [0, p k -1] × [0, n -1] : i + l = j} 2 ≤ 2n 2 (n + p k ) α 2 k ≤ 4n 2 1 p k k log k
where the last inequality holds as n ≤ p k . The random variables {S n (f k )} {k: p k ≥n} are independent and their sum is Z La (n), therefore

Z La (n) 2 2 = k: n≤p k S n (f k ) 2 2 ≤ 4n 2 k: n≤p k 1 p k k log k . Since in addition, p k ≤ 2 k , then k: n≤p k 1 p k k log k ≤ 1 log n k: n≤p k 1 p k ≤ 4 n log n .
and Z La (n) For a k with p k < n < d k we have,

S n (f k ) = n-1 j=0 p k -1 i=0 U i+j fk -U d k n-1 j=0 p k -1 i=0 U i+j fk hence Ŷ (n) = k:p k <n<d k S n (f k ) = k:p k <n<d k (A k + B k + C k ) -U d k (A k + B k + C k )
where

A k := p k -2 i=0 (i + 1)U i fk , B k := p k n-1 i=p k -1 U i fk , C k := n+p k -2 i=n (n + p k -1 -i) U i fk . Using independence of U i fk , 0 ≤ i ≤ p k -1, we deduce A k 2 2 ≤ p k -2 i=0 (i + 1) 2 1 p 2 k k log k ≤ p k k log k 2 k k By Lemma 18 we derive, k:p k <n<d k A k 2 2 ≤ k:p k <n<d k 2 k k k: √ log n≤k≤log n 2 k k = O n log n .
Similarly we derive

k:p k <n<d k C k 2 2 = O n log n .
Finally,

k:p k <n<d k B k 2 2 = k:p k <n<d k (n + 1 -p k ) α 2 k p 2 k ∼   n k:p k <n<d k 1 k log k - k:p k <n<d k p k k log k   . By Lemma 18, k:p k <n<d k p k k log k k: √ log n<k<log n 2 k k = O n log n . Since lim k→∞ log p k k = 1, then 1 k:p k <n<d k 1 k log k ∼ k: √ log n<k<log n 1 k log k ∼ log n √ log n dx x log x = (ln 2) 2 . 1 log n √ log n dx x log x = ln 2 ln ln(log x) -ln ln( √ log x) = (ln 2) 2
The random variables

A k , B k , C k , U d k A k , U d k B k , U d k C k , where the index k satisfies p k < n < d k are all independent, whence Ŷ (n) 2 2 n = 2 n k:p k <n<d k A k 2 2 + B k 2 2 + C k 2 2 = 2(ln 2) 2 + o(1),
showing that (8) holds.

Define for i ∈ {1, .., n} a function

Y i (n) : X → Z by Y i (n) := {k:p k ≤i+1, p k <n<d k } p k U i fk -U d k +i fk . Because n-1 i=1 Y i (n) = {k:p k <n<d k } B k -U d k B k ,
the following is deduced from Lemma 5 and the proof of Lemma 6.

Proposition 7. For every n ∈ N, the random variables S n (f ) -

n i=1 Y i (n) and n i=1 Y i (n) are independent and S n (f ) - n-1 i=1 Y i (n) 2 2 = O n √ log n . Proposition 8. S n (f ) converges in distribution to the normal law N (0, σ 2 ) with σ 2 = 2 (ln 2) 2 .
Proof. By Proposition 7 it is sufficient to prove the convergence for 1

√ n n i=1 Y i (n).
Since for every n ∈ N, the random variables Y 1 (n), ..., Y n (n) are independent and centered, this will follow once we verify the Lindeberg's condition

∀ > 0, 1 nσ 2 n-1 i=1 X Y i (n)1 [Yi(n) 2 > 2 σ 2 n] 2 dµ ----→ n→∞ 0. Because p k ≤ 2 k , if J ⊂ N is such that for all j ∈ J, 2 j < σ √ n 8 then j∈J p j U i fj -U dj +i fj ≤ 2 j∈J 2 j ≤ σ √ n 2 .
Therefore, writing A n, := k : log(σ )

-1 + 1 2 log n ≤ k ≤ log n , if |Y i (n)| > σ √ n then, |Y i (n)| ≤ 2 k∈An, p k U i fk -U d k +i fk .
We conclude that

Y i (n)1 [Yi(n) 2 > 2 σ 2 n] 2 ≤ 4   k∈An, p k U i fk -U d k +i fk   2 .
The terms p k U i fk which appear in the right hand side are mutually independent, thus for all i ∈ {1, .., n -1},

X Y i (n)1 [Yi(n) 2 > 2 σ 2 n] 2 dµ ≤ 4 X   k∈An, p k U i fk -U d k +i fk   2 dµ = 8 k∈An, p 2 k fk 2 2 = 8 k∈An, 1 k log k ln ln log(n) -ln ln log(σ ) -1 + 1 2 log n = o(1),
as n → ∞. This proves that the Lindeberg condition holds.

Proof of the local CLT

For the proof of the local CLT we first start with a new presentation of the main term n i=1 Y i (n). For this purpose let

I n := k ∈ 2N : 2 k < n < 2 k 2
and for k ∈ I n set

V k := n-1 i=2 k p k U i fk -U i+d k fk + p k+1 U i fk+1 -U i+d k+1 fk+1
To explain, I n roughly denotes the even integers in the segment p k < n < d k and for each k ∈ I n , V k is, up to an independent term with small second moment, almost equal to

B k + B k+1 -U d k (B k ) -U d k+1 (B k+1
). The next Proposition makes this claim precise. We use the notation

U n := k∈In V k , W n := n i=1 Y i (n) Proposition 9.
The random variables U n and E n := W n -U n are independent and

E n 2 2 = O n √ log n .
Proof. There are three types of terms appearing in E n . The first is if there exists an even integer k such that p k < n < d k and n ≤ p k+1 = 2 k + 1. Since for k even, p k = 2 k , this happens if and only if log(n -1) = k ∈ 2N. In this case, writing k = log(n -1), then V log(n-1) contains the term

n-1 i=2 k p k+1 U i fk+1 -U i+d k+1 fk+1 = n U n-1 flog(n-1)+1 -U n-1+d log(n-1)+1 flog(n-1)+1 = A(n),
where we have used that 2 k = n -1 and p k+1 = n. The second type is when there exists an even k such that d k ≤ n < d k+1 . In this case p k+1 < n < d k+1 , therefore B k+1 -U d k+1 B k+1 appears in W n and does not appear in U n .

Since n < d k+1 we see that log n < k + 1. This implies that

B k+1 2 2 = (n + 1 -p k+1 ) p 2 k+1 α 2 k+1 ≤ n k + 1 ≤ n log n . ( 9
)
Finally the third type of terms comes from the fact that for each k ∈ I n we are not including

p k U p k -1 fk -U d k +p k -1
fk in the definition of V k while it does appear in B k . We conclude that

E n = -1 [log(n-1)∈2N] A(n) + k∈In p k U p k -1 fk -U d k +p k -1 fk + 1 ∃k∈2N:[d k ≤n<d k+1 ] B k+1 -U d k+1 B k+1 .
The independence of E n and U n follows from properties (a) and (b) in the construction of the functions fk . Finally as the terms in the sum of E n are independent, using the bound on the last term (if and when it appears)

E n 2 2 = 2 n 2 1 [log(n-1)∈2N] flog(n-1)+1 2 2 + k∈In p 2 k fk 2 2 + O n √ log n ≤ 2 n 2 α log(n-1) 2 + k∈In p 2 k α 2 k + O n √ log n = o(1) + k: √ log(n)<k<log(n) 1 k log k + O n √ log n = O n √ log n .
Combining this with Proposition 7, we have shown.

Corollary 10. The random variables S n (f ) -U n and U n are independent and

S n (f ) -U n 2 2 = O n log(n) .
Consequently, 1

√ n U n converges in distribution to a normal law with variance σ 2 = 2(ln n) 2 . In the remaining part of this section we will prove that U n satisfies a local CLT and use Proposition 16 to deduce the local CLT for S n (f ).

Theorem 11. Writing σ 2 = 2(ln n) 2 then, sup x∈Z √ nµ (U n = x) - 1 √ 2πσ 2 e -x 2 /2nσ 2 ----→ n→∞ 0.
Deduction of Theorem 4. By Corollary 10, the random variables S n (f ) and U n satisfy the conditions of X n and Y n in Proposition 16. Thus by Theorem 11 and Proposition 16 we see that

sup x∈Z √ nµ (S n (f ) = x) - 1 √ 2πσ 2 e -x 2 /2nσ 2 ----→ n→∞ 0.
For the proof of Theorem 11 introduce the characteristic function of U n ,

φ n (t) := exp (itU n ) dµ.
The following two Lemmas are the core estimates which are used in the domination part, as in the proof of the local CLT in Terrence Tao's blog.

Lemma 12. There exists c > 0 such that for all

4 √ n ≤ |x| ≤ π √ n, φ n x √ n ≤ exp -c 4 √ n ≤ exp -d |x|
where d := c √ π . Lemma 13. There exists N ∈ N and a constant L > 0 such that for all n > N and |x| ≤ 4 √ n,

φ n x √ n ≤ exp -Lx 2 .
Proof of Theorem 11. Let m ∈ Z. By Fourier inversion,

µ (U n = m) = 1 2π π -π φ n (t)e -itm dt Applying the change of variable, t = x √ n , we see that √ nµ (U n = m) = 1 2π π √ n -π √ n φ n x √ n e -ixm/ √ n dx Since, 1 √ 2πσ e -m 2 /2nσ 2 = 1 2π R e -σ 2 x 2 /2 e -ixm/ √ n dx then it remains to show that sup m∈Z 1 2π π √ n -π √ n φ n x √ n e -ixm/ √ n dx - 1 2π R e -σ 2 x 2 /2 e -ixm/ √ n dx = o(1).
Using the triangle inequality and

|x|≥π √ n e -σ 2 x 2 /2 dx ----→ n→∞ 0, it suffices to show that π √ n -π √ n φ n x √ n -e -σ 2 x 2 /2 dx ----→ n→∞ 0. ( 10
)
Since Un √ n converges in distribution to a centered normal with variance σ 2 , it follows from Levy's continuity theorem that for all x ∈ R,

Ψ n (x) := 1 [-π √ n,π √ n ](x) φ n x √ n -e -σ 2 x 2 /2 ----→ n→∞ 0.
In addition, by Lemmas 12 and 13, for every n large, for all |x| ≤ π √ n,

Ψ n (x) ≤ G(x)
where

G(x) = e -σ 2 x 2 /2 + max exp -Lx 2 , exp -d |x| .
and L and d are the constants in the Lemmas. Since G is integrable, it follows from the dominated convergence theorem that

π √ n -π √ n φ n x √ n -e -σ 2 x 2 /2 dx = R Ψ n (x)dx ----→ n→∞ 0.
The proof is thus concluded.

Remark 14. For k ∈ I n and j ≤ n we let X k (j) be an i.i.d. sequence which is distributed as

p k U j fk + p k+1 U j fk+1 .
The random variable X k (j) takes values in 0, ±1, ±2 k , ± 2 k + 1 , ± 2 k+1 + 1 . We assume {X k (j)} k∈In, j≤n are independent. Note that writing

X n := k∈In n j=2 k X k (j), then if X n , X n are independent identically distributed, then U n d = X n + X n . Therefore φ n (t) = k∈In [E (exp(itX k (1)))] 2(n-2 k )
Proof of Lemma 12. First note that for all k ∈ I n

P (X k (1) = z) =            1 -α 2 k 1 -α 2 k+1 , z = 0 α 2 k 2 1 -α 2 k+1 z ∈ ±2 k α 2 k+1 2 1 -α 2 k z ∈ ± 2 k + 1 (α k α k+1 ) 2 4 z ∈ ±1, ± 2 k+1 + 1 so substituting t = x √ n ∈ [-π, π], φ n x √ n = k∈In E exp(ixX k (1)/ √ n 2(n-2 k ) = k∈In |E (exp(itX k (1))| 2(n-2 k ) ≤ k∈In 1 - (α k α k+1 ) 2 2 (1 -cos t) 2(n-2 k ) .
For the bound of 4 √ n ≤ |x| ≤ π √ n note that for such x,

cos (t) ≤ cos 1 4 √ n ≤ 1 - 1 4 √ n .
This shows that for 4

√ n ≤ |x| ≤ π √ n, φ n x √ n ≤ k∈In 1 - (α k α k+1 ) 2 8 √ n 2(n-2 k ) ≤ exp - 1 4 √ n k∈In n (α k α k+1 ) 2 - k∈In 2 k α 4 k .
A calculation shows that where d = c √ π . Define for n ∈ N, Proof. Now Υ n (j), 1 ≤ j ≤ n are independent, for all 1 ≤ j ≤ n, E (Υ n (j)) = 0 and

k∈In 2 k α 4 k ≤ k∈In 2 k 2 -4k = o(1), as n → ∞,
J n := k ∈ N : 2 k ≤ 4 √ n 3 , n < 2 k 2 = log n, log n 4 -log 3 ∩ N.
E (Υ n (j)) 2 = {k∈Jn:2 k ≤j} p 2 k α 2 k = {k∈Jn:2 k ≤j} 1 k log k . For all n 2 ≤ j ≤ n, 2 {k∈Jn:2 k ≤j} 1 k log k = k∈Jn 1 k log k . ( 11
)
A similar argument as in the proof of Lemma 6 shows that there exists β > 0 such that

k∈Jn 1 k log k = √ log n<k≤ log n 4 -log 3 1 k log k ∼ log(n) 4 -log 3 √ log(n) dx x log x = ln 2 ln ln log n 4 -log 3 -ln ln log n = ln 2 ln ln(log n) -ln ln(log(n)) 2 + o(1) = (ln 2) 2 + o(1).
Consequently, for all n 2 ≤ j ≤ n,

E (Υ n (j)) 2 ∼ (ln 2) 2 , as n → ∞, (12)
Note that the latter asymptotic equivalence is uniform when n → ∞ and n 2 ≤ j ≤ n. Since for all k ∈ J n ,

p k ≤ 2 k ≤ 4 √ n 3 , for all k ∈ J n and 1 ≤ j ≤ n, E p k U j fk 4 = p 4 k α 2 k = p 2 k k log k ≤ √ n 9 1 k log k .
By this, the independence of p k U j fk k∈Jn and that for all k ∈ J n and j ∈ N, E p k U j fk = 0 we see that for all n 2 ≤ j ≤ n,

E (Υ n (j)) 4 = E   k∈Jn p k U j fk 4   ≤ k∈Jn E p k U j fk 4 + k∈Jn E p k U j fk 2 2 ≤ √ n 9 k∈Jn 1 k log k + E (Υ n (j)) 2 2 = √ n 9 (ln 2) 2 + o √ n .
There exists N such that for all n > N , and

n 2 ≤ j ≤ n E (Υ n (j)) 2 ≥ (ln 2) 2 2 ( 13
)
and in addition

E (Υ n (j)) 4 ≤ 2(ln 2) 2 9 √ n ≤ 4 √ n 9 E (Υ n (j)) 2 .
Furthermore, U i fk {k∈Jn: 2 k ≤j} is a sequence of independent and symmetric random variables, thus

E (Υ n (j)) = E (Υ n (j)) 3 = 0.
It follows that for all n/2 ≤ j ≤ n, 3 and |t| ≤ 1 The conclusion follows with L = (ln 2) 2 72 . Proof of Lemma 13. Let N and L be as in Lemma 15. Since Z n and U n -Z n are independent, then for all n > N , and |x| ≤ 4 √ n,

√ n |E (exp (itΥ n (j)))| ≤ 1 - E (Υ n (j)) 2 2 t 2 + t 4 4! E (Υ n (j)) 4 ≤ 1 -E (Υ n (j)) 2 t 2 1 2 - 4 √ n 9 t 2 √ nt 2 ≤
E exp(ixU n / √ n = E exp(ix (U n -Z n ) / √ n E exp(ixZ n / √ n ≤ E exp(ixZ n / √ n ≤ exp -Lx 2 )

Appendix

The first result in the appendix is that the local limit theorem persists under addition of small independent noise. These type of arguments and statements are not new. We include a statement which is especially tailored for our construction. In order to simplify the notation in the proof, we would make use of the following reformulation of (15): There exists r : N → [0, ∞) such that r(n) √ n → 0 and for all x ∈ Z,

P (Y n = x) = 1 √ 2πnσ 2 e -x 2 2nσ 2 ± r(n).
In the course of the proof, the function r(n) will denote a o 1 √ n function which may change from line to line.

Proof. By changing from X n to 1 σ X n we can and will assume that σ 2 = 1. Write a(n) = We split the sum into |z| ≤ a n and |z| > a n . A consequence of ( 15) is that there exists C > 0 such that for all y ∈ Z and n ∈ N,

P (Y n = y) ≤ C √ n .
And, Using these rather trivial bounds and the reformulation of [START_REF] Lacey | On central limit theorems, modulus of continuity and Diophantine type for irrational rotations[END_REF] we conclude that The conclusion follows from the latter asymptotic equality and [START_REF] Liardet | Sums of continuous and differentiable functions in dynamical systems[END_REF].

The following estimate is used in bounding the L 2 norm of the first term in Proposition 7.

Lemma 17. There exists a constanst K > 0 such that for all n ∈ N,

k∈N: 2≤k≤ √ log n 2 k 2 k log k ≤ K n log(n)
.

Lemma 18.

k:2 k ≤n

2 k k = O n log n .
Proof of Lemmas 18 and 18. Both results follow from the following reasoning. If (a k ) ∞ k=1 is a sequence of positive reals such that there exists q > 1 for which for all n ∈ N, a n+1 /a n ≥ q then for all n ∈ N,

n k=1 a k ≤ a n ∞ k=1 q -k ≤ a n 1 -q .

  ) Let Y be a rung of the castle. Since the castle is ξ-independent, m(D ∩ Y ) = m(D)m(Y ) and D ∩ Y is a finite union of sets of the form T -r C with C ∈ ζ 1 and r a fixed integer, 0 ≤ r ≤ 2l -1, 2l. (depending on the tower of the castle to which Y belongs). Denote C = T -r C.

  is a triangular array. However, Proposition 2 enables us to get property (b) which is more than a realization of this triangular array. This step is crucial in what follows. Let us define f := ∞ k=1 f k where for each k ∈ N,

2 2 =

 2 O n log n . This together with (7) implies part (a). Part (b) is a direct consequence of part (a). Lemma 6. lim n→∞ 1 n S n (f )

2 2

 2 = 2(ln 2) 2 =: σ 2 Proof. By Lemma 5.(b) it is enough to show that lim n→∞

10 √- 5 √ log n 4 √

 1054 log n/2 = C2 -5 √ log n for some global constant C which does not depend on n. Since √ n2 n we have shown that there exists c > 0 such that for all 4 √ n ≤ x ≤ π √ n, |φ n (x)| ≤ exp -c 4 √ n ≤ exp -d |x| .

and for 1 Lemma 15 .

 115 ≤ j ≤ n, Υ n (j) = k∈Jn: 2 k ≤j p k U j fk and finally Z n = n j=1 Υ n (j). Note that {Υ n (j)}n j=1 are independent and that U n -Z n and Z n are independent by the construction. There exists N ∈ N and a constant L > 0 such that for all n > N and |x| ≤ 4 √ n, E exp(ixZ n / √ n ≤ exp -Lx 2

4

 4

Proposition 16 .

 16 Suppose that X n = Y n + Z n which are for each n ∈ N, Y n and Z n are Z-valued independent random variables. If sup x∈Z √ nP (Y n = x) -

2 . 4 √

 24 By Markov inequality,P (|Z n | ≥ a(n)) ≤ D 4 log(n) ,where D is any constant such that sup n∈N Fix x ∈ Z and note that as Y n and Z n are independent,P (X n = x) = z∈Z P (Y n = x -z) P (Z n = z) .

  z∈Z\[-a(n),a(n)]P (Y n = x -z) P (Z n = z) ≤ C √ n P (|Z n | ≥ a(n)) ≤ CD √ n 4 log(n) . (16)If |z| ≤ a(n) and |x| > √ n log 1/9 (n) then for all large n, uniformly in z = ±a(n),The last equality up to a o(1) term follows fromx 2 /n ≥ log 2/9 (n) → ∞ as n → ∞. If |x| ≤ √ n log 1/9 (n) then |xz| ≤

√ 2πn exp - x 2

 2 z∈Z∩[-a(n),a(n)]P (Y n = x -z) P (Z n = z) = z∈Z∩[-a(n),a(n)] exp -(x-z) 2 2n ) √ 2πn P (Z n = z) ± r(n) = z∈Z∩[-a(n),a(n)] Z n = z) ± r(n) = P (|Z n | ≤ a n ) 1 2n ± (o(1) + r(n)) As P (|Z n | ≤ a n ) = 1 + o(1), z∈Z∩[-a(n),a(n)] P (Y n = x -z) P (Z n = z)

For all k ∈ Jn, 2 k ≤ 4 √ n/3 < n/2

See for example[START_REF] Durrett | Probability: theory and examples[END_REF][pp.[101][102][103]

Einstein Institute of Mathematics, Hebrew University of Jerusalem, Edmond J. Safra Campus, Jerusalem 91904, IsraelEmail address: zemer.kosloff@mail.huji.ac.il Laboratoire de Mathématiques Raphael Salem, UMR 6085, Université de Rouen, Normandie, FranceEmail address: dalibor.volny@univ-rouen.fr

The research of Z.K. was partially supported by ISF grant No. 1570/17.