LOCAL LIMIT THEOREM IN DETERMINISTIC SYSTEMS

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LOCAL LIMIT THEOREM IN DETERMINISTIC SYSTEMS

Zemer Kosloff, Dalibor Volny

To cite this version:

Zemer Kosloff, Dalibor Volny. LOCAL LIMIT THEOREM IN DETERMINISTIC SYSTEMS. 2019.

�hal-02386881�

ZEMER KOSLOFF AND DALIBOR VOLNY

Dedicated to Manfred Denker whose work is an inspiration for us.

Abstract. We show that for every ergodic and aperiodic probability preserving system, there exists a Zvalued, square integrable functionf such that the partial sums process of the time series

f◦T^{i ∞}

i=0satisfies the lattice local limit theorem.

1. 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 processSn(f) :=Pn−1

k=0f◦T^k satisfies the lattice local central limit theorem.

A centered function f :X →Rsatisfies the central limit theorem if for all u∈R m

Sn(f) kSn(f)k₂ ≤u

−−−−→

n→∞

Z u

−∞

e^−x

2 2 dx.

A function f :X →ZwithR

f dm= 0 such that lim_n→∞^kSn(f)k_n ²² =σ²>0 satisfies a lattice local central limit theorem if

sup

x∈Z

√n·m(Sn(f) =x)−e^−x2/(^2nσ2)

√ 2πσ²

−−−−→

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 iff satisfies a local central limit theorem (whether lattice or non-lattice) we will say thatf 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◦Tⁿ}^∞_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 [17] existence of CLT functions such that the variance ofSn(f) grows linearly withn. Some years later, the seminal paper of Burton and Denker [4] showed that for every aperiodic dynamical system there exists a function f which satisfies the central limit theorem. By [20], the set of CLT functions is a meagre set inL²₀:=

f ∈L²(m) : R

f dm= 0 . See also [16] and [7] for such results in other function spaces.

Following [4], several extensions and improvements regarding existence and bounds of the regularity of CLT functions were done, see for example [12],[14],[15],[6]. The most relevant to this work is [21] 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 [3],[19].

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 [18],[11],[1],[2] and

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

1

[9].

The methods of proving the central limit theorem in [4] and [21] 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 preserv- ing dynamical system (X,B, m, T) there exists a square integrable function f : X → Z with R f dm= 0which satisfies the lattice local central limit theorem.

The construction of the aforementioned function relies on a new version of the stochastic coding theorem [10],[13]. 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 meansae^−b≤x≤xe^b.

For sequences a(n), b(n)>0 we will writea(n)∼b(n) if lim_n→∞^a(n)_b(n) = 1. In some cases it will be denoted with the littleonotation, that isa(n) =b(n) +o(1).

By a(n) =O(b(n)) anda(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.

2. The strong Alpern tower lemma and realizations of triangular arrays Let Abe a finite set, {dn}^∞_n=1 an integer valued sequence and Xn,m, n ∈N, 1≤m≤dn

be anA-valued triangular array. In our setting this means

• For everyn∈Nandm∈ {0, ..., dn−1}, Xn,m: (Ω,F,P)→A.

• For everyn∈N,Xn,1, . . . , Xn,d_n are identically distributed.

• {Xn,m}_n∈

N,m∈{1,..,d_n}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 [10], [13] for results in a similar flavor.

Proposition 1. Let(X,B, µ, T)be an ergodic invertible probability measure preserving trans- formation. For every finite setA, and anAvalued triangular arrayX:={Xn,m}_{n∈N,m∈{0,..,d}

n}, there exists a sequence of functions f_n :X → A such that {fn◦T^m}_{n∈N,m∈{0,..,d}

n−1} and X have the same distribution.

The construction of the functionsfn is by induction onN ∈Nso that

{fn◦T^m}1≤n≤N,m∈{0,..,dn−1} and{Xn,m}1≤n≤N,m∈{1,..,dn} have the same distribution. For a fixedN one can considerξ, the finite partition ofX according to the value of the vector valued function

(GN)_n,m(x) :=fn◦T^m(x), 1≤n≤N, 0≤m≤dn−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 trans- formation and ξa finite measurable partition ofX. For every finite setAandX₁, X₂, .., 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{Xi}^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. GivenN ∈N, a{N, N+ 1}Alpern-Rokhlin castlefor (X,B, m, T) is given by two measurable setsB_N, B_N+1 such that

•

T^jB_L: L∈ {N, N + 1}, 0≤j < L is a partition of (X,B, m) to pairwise disjoint sets..

• We callBN, BN+1thebase elementsof the castle and the atoms in the corresponding partition are referred to asrungs. We callT^NBn]T^N+1BN+1 thetop 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 [5][Corollary 1], there exists a ξ-independent{2l,2l+ 1}castle with basesB =B_2l andF =B_2l+1. Note that asT is invertible then,

T T^2l−1B]T^2lF

=B]F.

Letζ1be the partition of the top of the tower{T^2l−1B, T^2lF}which is its refinement according to ∨^2l_i=0Tⁱξ. First we definef :∪

T^−jC: C∈ ζ1,0≤j < l →Aas follows. Partition each C∈ζ₁ toA^l elements

C_a: a∈A^l such that for everya= (a_i)^l−1_i=0∈A^l, m(Ca) =m(C)

l−1

Y

i=0

P(X_i+1=a_i).

and setf(T^−l+1+ix) =a_i,i= 0, . . . , l−1, if there existsa∈A^l andC∈ζ₁ such thatx∈Ca. It remains to definef in the bottom rungs of the tower. Byζ₁⁰ we denote the refinement of ζ1 by the setsCa. Byζ we denote the joint partition of the base of the castle byT^−2l+1(ζ₁⁰ ∩ T^2l−1B),T(ζ₁⁰ ∩T^2l−1B),T^−2l(ζ₁⁰ ∩T^2lF),T(ζ₁⁰ ∩T^2lF).

For everyC∈ζwe do as follows. IfC⊂Bthen we partitionCtoA^lelements

Ca: a∈A^l such that for every a= (a_i)^l−1_i=0∈A^l,

m(C_a) =m(C)

l−1

Y

i=0

P(X_i+1=a_i) and set f(Tⁱx) =a_i if there exists a∈A^landC∈ζ such thatx∈C_a.

IfC⊂ζ∩F we do the same withl+ 1 replacingl.

For anyC∈ζ₁ we thus have m (T^−l+1C)∩ ∩^l−1_i=0

f◦Tⁱ=ai+1

=m(T^−l+1C)

l

Y

i=1

P(Xi=ai), forC∈ζ∩B we have

m C∩ ∩^l−1_i=0

f ◦Tⁱ=ai+1

=m(C)

l

Y

i=1

P(Xi=ai), and forC∈ζ∩F we have

m C∩ ∩^l_i=0

f ◦Tⁱ=a_i+1

=m(C)

l+1

Y

i=1

P(X_i=a_i).

Moreover, forC∈ζ₁∩T^2l−1B the setsT^−2l+1Caare independent of (f◦Tⁱ)^l−1_i=0(conditionally onT^−2l+1C) hence

m (T^−2l+1C)∩ ∩^2l−1_i=0

f ◦Tⁱ=ai+1

=m(C)

2l

Y

i=1

P(Xi=ai) (1) and forC∈ζ1 withT^−2lC⊂F we have

m (T^−2lC)∩ ∩^2l_i=0

f◦Tⁱ=a_i+1

=m(C)

2l+1

Y

i=1

P(X_i=a_i). (2) LetC∈T^−2l+1(ζ₁∩T^2l−1B) orC∈T^−2l(ζ₁∩T^2lF), 0≤k≤l. We have

T^kC∩ ∩^2l−1−k_i=0

f ◦Tⁱ=ai+k+1

=T^k C∩ ∩^2l−1_i=k

f◦Tⁱ=ai+1

;

by (1) and (2) we get

m (T^kC)∩ ∩^2l−1−k_i=0

f ◦Tⁱ=ai+k+1

=m(T^kC)

2l

Y

i=k+1

P(Xi=ai). (3) Let us show a similar equation for T^−l+1C, C∈ζ₁.

We have

T^l (T^−l+1C_a)∩(T^−lB))∩ ∩^2l−1_i=l

f◦Tⁱ=a_i+1

= (B∩T C_a)∩ ∩^l−1_i=0

f◦Tⁱ=a_i+l+1 and the same equality holds forF, hence

m (T^−l+1C_a)∩(T^−lB))∩ ∩^2l−1_i=l

f ◦Tⁱ=a_i+1

= m (B∩T Ca)∩ ∩^l−1_i=0

f◦Tⁱ=ai+l+1 and

m (T^−l+1Ca)∩(T^−lF))∩ ∩^2l−1_i=l

f◦Tⁱ=ai+1

= m (F∩T Ca)∩ ∩^l−1_i=0

f◦Tⁱ=ai+l+1

. Therefore,

m (T^−l+1Ca)∩ ∩^2l−1_i=l

f◦Tⁱ=ai+1

=m (T Ca)∩ ∩^l−1_i=0

f◦Tⁱ=ai+l+1

and by independence of ζandf, . . . , f◦T^l−1onB∪F we deduce

m (T^−l+1Ca)∩ ∩^2l−1_i=l

f◦Tⁱ=ai+1

=m(T Ca)m (B∪F)∩ ∩^l−1_i=0

f◦Tⁱ=ai+l+1

=m(C_a)

l

Y

i=1

P(X_i=a_i+l+1)

=m(C)

2l

Y

i=1

P(Xi=ai). Recall that if C∈ζ₁ anda= (a_i)^l_i=1∈A^l then

T^−l+1Ca=T^−l+1C∩ ∩^l−1_i=0

f◦Tⁱ =ai+1 . Therefore,

m (T^−l+1C)∩ ∩^2l−1_i=0

f◦Tⁱ=ai+1

=m (T^−l+1Ca)∩ ∩^2l−1_i=l

f◦Tⁱ=ai+1

=m(C)

2l

Y

i=1

P(X_i=a_i), hence

m (T^−l+1C)∩ ∩^2l−1_i=0

f ◦Tⁱ=ai+1

=m((T^−l+1C))

2l

Y

i=1

P(Xi=ai). For 0≤k≤l−1 we thus have

m (T^−l+k+1C)∩ ∩^2l−1_i=0

f◦Tⁱ=ai+1

=m((T^−l+1C))

2l

Y

i=1

P(Xi=ai). (4) We show thatf satisfies the conclusion of the proposition. Let D∈ξ anda∈A^l. We claim that

m D∩ ∩^l−1_i=0

f ◦Tⁱ=ai+1

=m(D)

l

Y

i=1

P(Xi=ai). (5) 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^−rC with C ∈ ζ₁ and r a fixed integer, 0 ≤ r ≤ 2l−1,2l. (depending on the tower of the castle to which Y belongs). Denote C⁰=T^−rC.

From (3) and (4) it follows that m C⁰∩ ∩^l−1_i=0

f◦Tⁱ =a_i+1

=m(C)

l

Y

i=1

P(X_i=a_i). (6) Because ζ1 is finer than (T^2l−1B∪T^2lF)∩ ∨^2l_i=0Tⁱξ, D∩Y is a union of setsC⁰. D is just a disjoint union of the sets C⁰ over all rungs Y. Summing equations (6) for all suchC⁰ we see that equation (5) holds.

3. Definition of the function and proof of the CLT

Let (X,B, µ, T) be an ergodic invertible probability measure preserving transformation and U : L2(X, µ) → L2(X, µ) is its corresponding Koopman operator. In this susbsection we construct the functionf for which the local limit theorem holds.

Fork∈Nwe define

p_k:=

(2^k, k even 2^k−1+ 1, k odd.

and

dk:= 2^k2, and αk:= 1 p_k√

klogk.

By a repeated inductive iteration of Proposition 2, there exists a sequence of functions ¯fk : X → {−1,0,1}such that:

(a) For everyk∈N, f¯_k◦T^{j 2dk+pk}

j=0 is an i.i.d. sequence,{−1,0,1}valued and µ f¯k= 1

=µ f¯k=−1

= α_k² 2 .

(b) For everyk≥2, the finite sequence f¯k◦T^{j 2d}_j=0^k+pk is independent of Ak:=f¯l◦T^j: 1≤l < k, 0≤j≤2dk+pk .

To explain how we apply the proposition, the values of the functionsg∈ Ak induce a partition ξ of X into 3^2dk+pk elements. Write X₁, X₂, X₃, ..., X_2dk+pk for an i.i.d. sequence such that X1 is {−1,0,1} distributed withE(X1) = 0 and E X₁²

=α²_k and apply Proposition 2. In this definition the sequence of functions

f¯_k◦T^j :k∈N, 0≤j ≤2d_k+p_k

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 :=P∞

k=1fk where for eachk∈N, fk :=

p_k−1

X

i=0

Uⁱf¯k−U^dk

p_k−1

X

i=0

Uⁱf¯k

! .

It is worth to note that each of the functionf_k is a coboundary with transfer function gk :=

d_k−1

X

j=0 pk−1

X

i=0

U^i+jf¯k, fk=gk−U gk. Proposition 3. f ∈L²(X, µ).

Proof. Fix k ∈ N and write Vk,i := Uⁱf¯k −U^dk+if¯k where i ∈ {0,1, ..., pk −1}. This is a sequence of i.i.d. random variables withR

XVk,idµ= 0 andkVk,ik²₂= 2α²_k. Therefore kfkk²₂=

Z

X pk−1

X

i=0

Vk,i

!² dm

=

p_k−1

X

i=0

Z

X

(Vk,i)²dm= 2α²_kpk≤ 2 pk

.

By condition (b) in the definition of the ¯f_k’s, the functionsf₁, f₂, ..are independent. As they are also centered,

kfk²₂=

∞

X

k=1

kfkk²₂≤

∞

X

k=1

2 p_k <∞.

We conclude thatf is well defined.

Theorem 4. The function f : X → R satisfies the local limit theorem with σ² := 2(ln 2)². That is

sup

x∈Z

√nµ(Sn(f) =x)−e^−x2/(^2nσ2)

√ 2πσ²

−−−−→

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 dk

instead of an exponential sequence as in [21] is used in this step.

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

In Section 4, we show the local limit theorem forYn. 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 ofY_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

nas in Lemma 13.

3.1. Proof of the CLT. We start by presentingSn(f) as a sum of three terms depending on the scale ofkwith respect ton. That is

Sn(f) =ZSm(n) + ˆY(n) +ZLa(n).

where

ZSm(n) := X

k:dk≤n

Sn(fk) Yˆ(n) := X

k:pk<n<dk

Sn(fk) ZLa(n) := X

k:n≤p_k

Sn(fk)

Lemma 5. For everyn∈Nthe random variablesZSm(n),Yˆ(n), ZLa(n)are independent and (a) kZSm(n) +ZLa(n)k²₂=O

√n logn

. (b) ^√1_nYˆ(n)−^√1_nSn(f)−−−−→

n→∞ 0 inL²(X, µ).

Proof. For each k ∈N, S_n(f_k) is a sum of functions from the sequencef¯_k◦T^{i n+d}_i=0^k+pk−1. For allk’s appearing in the sums describing ˆY(n) andZLa(n), one hasn < dk, therefore ˆY(n) andZLa(n) are sums of functions of the formf¯k◦T^{i 2d}_i=0^k+pk withk >√

lognandZSm(n) is

a sum of functions fromf¯k◦T^{i 2d}_i=0^k∗+pk∗ withk≤√

lognandk^∗ is the smallest integer such that k^∗>√

logn.

By property (b) in the definition of the ¯fk’s we see that ZSm(n) is independent of ˆY(n) and ZLa(n). A similar reasonning using property (b) of the functions ¯fkshows thatZSm(n),Yˆ(n), ZLa(n) are independent.

We now turn to prove part (a). Sincefk=gk−U gk then ZSm(n) = X

k:d_k≤n

(gk−Uⁿgk).

By independence of{f¯k◦Tⁱ}^p_i=0^k+dk, kgkk²₂=

dk+pk−1

X

j=0

(#{(l, i)∈[0, dk)×[0, pk) : l+i=j})² f¯k

2 2

< α²_kp²_k(pk+dk)≤ 2^k2+1 klogk.

The functions {gk}^∞_k=1are centered and independent, thus

X

k: dk≤n

gk

2

2

= X

k:dk≤n

kgkk²₂

≤2 X

k:d_k≤n

2^k2 klogk

= 2 X

k:k≤√

logn

2^k2 klogk =O

n

√logn

,

where the last inequality follows from Lemma 17. As U is unitary it follows that kZSm(n)k²₂=O

n

√logn

. (7)

It remains to bound kZLa(n)k²₂. First note that for k such that n≤pk, by independence of the summands

kSn(f_k)k²₂= 2

p_k−1+n

X

j=0

f¯_k

2#{(i, l)∈[0, p_k−1]×[0, n−1] :i+l=j}²

≤2n²(n+p_k)α²_k≤4n² 1

p_kklogk

where the last inequality holds asn≤p_k. The random variables{Sn(f_k)}_{k:p

k≥n} are inde- pendent and their sum isZ_La(n), therefore

kZ_La(n)k²₂= X

k:n≤pk

kS_n(f_k)k²₂≤4n² X

k:n≤pk

1 pkklogk. Since in addition,p_k ≤2^k, then

X

k:n≤pk

1

pkklogk ≤ 1 logn

X

k:n≤pk

1 pk

≤ 4 nlogn. and kZ_La(n)k²₂ = O

n logn

. This together with (7) implies part (a). Part (b) is a direct

consequence of part (a).

Lemma 6. lim_n→∞

1

nkS_n(f)k²₂

= 2(ln 2)²=:σ²

Proof. By Lemma 5.(b) it is enough to show that

n→∞lim

Yˆ(n)

2 2

n = 2(ln 2)². (8)

For a kwithpk < n < dk we have, S_n(f_k) =

n−1

X

j=0 p_k−1

X

i=0

U^i+jf¯_k−U^dk

n−1

X

j=0 p_k−1

X

i=0

U^i+jf¯_k hence

Yˆ(n) = X

k:p_k<n<d_k

Sn(fk) = X

k:p_k<n<d_k

(Ak+Bk+Ck)−U^dk(Ak+Bk+Ck) where

Ak:=

pk−2

X

i=0

(i+ 1)Uⁱf¯k, Bk :=pk n−1

X

i=pk−1

Uⁱf¯k, Ck:=

n+pk−2

X

i=n

(n+pk−1−i)Uⁱf¯k. Using independence ofUⁱf¯_k, 0≤i≤p_k−1, we deduce

kAkk²₂≤

pk−2

X

i=0

(i+ 1)² 1

p²_kklogk ≤ pk

klogk .2^k k By Lemma 18 we derive,

X

k:pk<n<dk

kAkk²₂≤ X

k:pk<n<dk

2^k k

. X

k:√

logn≤k≤logn

2^k k =O

n logn

.

Similarly we derive

X

k:p_k<n<d_k

kCkk²₂=O n

logn

. Finally,

X

k:p_k<n<d_k

kBkk²₂= X

k:p_k<n<d_k

(n+ 1−pk)α²_kp²_k

∼



n X

k:p_k<n<d_k

1

klogk− X

k:p_k<n<d_k

p_k klogk



. By Lemma 18,

X

k:p_k<n<d_k

pk

klogk . X

k:√

logn<k<logn

2^k k =O

n logn

.

Since limk→∞logp_k

k = 1, then¹ X

k:pk<n<dk

1

klogk ∼ X

k:√

logn<k<logn

1 klogk

∼ Z _logn

√logn

dx

xlogx = (ln 2)².

1Rlogn

√logn

dx

xlogx = ln 2 ln ln(logx)−ln ln(√ logx)

= (ln 2)²

The random variablesAk, Bk, Ck, U^dkAk, U^dkBk, U^dkCk, where the indexksatisfiespk < n <

dk are all independent, whence

Yˆ(n)

2 2

n = 2

n X

k:p_k<n<d_k

kAkk²₂+kBkk²₂+kCkk²₂

= 2(ln 2)²+o(1),

showing that (8) holds.

Define fori∈ {1, .., n} a functionYi(n) :X→Zby

Yi(n) := X

{k:pk≤i+1, pk<n<d_k}

pk Uⁱf¯k−U^dk+if¯k

.

Because

n−1

X

i=1

Yi(n) = X

{k:pk<n<dk}

Bk−U^dkBk , the following is deduced from Lemma 5 and the proof of Lemma 6.

Proposition 7. For every n∈N, the random variablesS_n(f)−Pn

i=1Y_i(n)and Pn i=1Y_i(n) are independent and

Sn(f)−

n−1

X

i=1

Yi(n)

2

2

=O n

√logn

.

Proposition 8. S_n(f)converges in distribution to the normal lawN(0, σ²)withσ²= 2 (ln 2)². Proof. By Proposition 7 it is sufficient to prove the convergence for ^√1_nPn

i=1Y_i(n). Since for every n ∈ N, the random variables Y1(n), ..., Yn(n) are independent and centered, this will follow once we verify the Lindeberg’s condition

∀ >0, 1 nσ²

n−1

X

i=1

Z

X

Yi(n)1[Y_i(n)²>²σ²n]

2

dµ−−−−→

n→∞ 0.

Because pk ≤2^k, ifJ ⊂Nis such that for allj∈J, 2^j< σ

√n 8 then

X

j∈J

pj Uⁱf¯j−U^dj+if¯j

≤2X

j∈J

2^j≤σ

√n 2 . Therefore, writing A_n,:=

k: log(σ)−1 + ¹₂logn≤k≤logn , if|Yi(n)|> σ√ nthen,

|Yi(n)| ≤2

X

k∈An,

pk Uⁱf¯k−U^dk+if¯k .

We conclude that

Yi(n)1_[Yi(n)2>²σ²n]

2

≤4



 X

k∈An,

pk Uⁱf¯k−U^dk+if¯k





2

.

The terms pkUⁱf¯k which appear in the right hand side are mutually independent, thus for all i∈ {1, .., n−1},

Z

X

Y_i(n)1_[Yi(n)2>²σ²n]

² dµ≤4

Z

X



 X

k∈An,

p_k Uⁱf¯_k−U^dk+if¯_k





2

dµ

= 8 X

k∈An,

p²_k f¯k

2 2

= 8 X

k∈An,

1 klogk

.ln ln log(n)−ln ln

log(σ)−1 + 1 2logn

=o(1), as n→ ∞. This proves that the Lindeberg condition holds.

4. Proof of the local CLT

For the proof of the local CLT we first start with a new presentation of the main term Pn

i=1Yi(n). For this purpose let In :=n

k∈2N: 2^k < n <2^k2o and fork∈In set

V_k:=

n−1

X

i=2^k

p_k Uⁱf¯_k−U^i+dkf¯_k

+p_k+1 Uⁱf¯_k+1−U^i+dk+1f¯_k+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 toB_k+ B_k+1−U^dk(B_k)−U^dk+1(B_k+1). The next Proposition makes this claim precise. We use the notation

Un:= X

k∈In

Vk, Wn:=

n

X

i=1

Yi(n)

Proposition 9. The random variablesUn andEn:=Wn−Un are independent and kEnk²₂=O

n

√logn

.

Proof. There are three types of terms appearing in En. The first is if there exists an even integer k such that pk < n < dk and n ≤ pk+1 = 2^k+ 1. Since for k even, pk = 2^k, this happens if and only if log(n−1) =k∈2N. In this case, writingk= log(n−1), thenV_log(n−1) contains the term

n−1

X

i=2^k

pk+1 Uⁱf¯k+1−U^i+dk+1f¯k+1

=n Uⁿ⁻¹f¯_log(n−1)+1−U^{n−1+dlog(n−1)+1}f¯_log(n−1)+1

=A(n), where we have used that 2^k =n−1 andp_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^dk+1B_k+1 appears in W_n and does not appear in U_n. Sincen < dk+1 we see that

plogn < k+ 1.

This implies that

kB_k+1k²₂= (n+ 1−p_k+1)p²_k+1α²_k+1≤ n

k+ 1 ≤ n

logn. (9)

Finally the third type of terms comes from the fact that for each k∈In we are not including pk U^pk−1f¯k−U^dk+pk−1f¯k

in the definition ofVk while it does appear inBk. We conclude that

En=−1[log(n−1)∈2N]A(n) +X

k∈In

pk U^pk−1f¯k−U^dk+pk−1f¯k

+ 1_{∃k∈2N:[dk≤n<dk+1]} Bk+1−U^dk+1Bk+1 .

The independence ofE_n andU_n follows from properties (a) and (b) in the construction of the functions ¯fk. Finally as the terms in the sum ofEn are independent, using the bound on the last term (if and when it appears)

kEnk²₂= 2 n²1[log(n−1)∈2N]

f¯_log(n−1)+1

2 2+ X

k∈In

p²_k f¯k

2 2

! +O

n

√logn

≤2 n² α_log(n−1)²

+ X

k∈In

p²_kα²_k

! +O

n

√logn

=o(1) + X

k: √

log(n)<k<log(n)

1 klogk +O

n

√logn

=O n

√logn

.

Combining this with Proposition 7, we have shown.

Corollary 10. The random variablesSn(f)−Un andUn are independent and kSn(f)−Unk²₂=O n

plog(n)

! .

Consequently, ^√1_nUn converges in distribution to a normal law with varianceσ²= 2(lnn)². In the remaining part of this section we will prove that Un satisfies a local CLT and use Proposition 16 to deduce the local CLT forSn(f).

Theorem 11. Writing σ²= 2(lnn)² then, sup

x∈Z

√nµ(Un=x)− 1

√

2πσ²e^−x2/2nσ2

−−−−→

n→∞ 0.

Deduction of Theorem 4. By Corollary 10, the random variables Sn(f) and Un satisfy the conditions of Xn and Yn in Proposition 16. Thus by Theorem 11 and Proposition 16 we see that

sup

x∈Z

√nµ(S_n(f) =x)− 1

√

2πσ²e^−x2/2nσ2

−−−−→

n→∞ 0.

For the proof of Theorem 11 introduce the characteristic function of Un,

φn(t) :=

Z

exp (itUn)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 existsc >0 such that for all √⁴

n≤ |x| ≤π√ n,

φ_n x

√n

≤exp −c√⁴ n

≤exp

−dp

|x|

where d:=^√c_π.

Lemma 13. There existsN ∈Nand a constantL >0such that for alln > N and|x| ≤ √⁴ n,

φ_n x

√n

≤exp −Lx² . Proof of Theorem 11. Letm∈Z. By Fourier inversion,

µ(U_n=m) = 1 2π

Z π

−π

φ_n(t)e^−itmdt Applying the change of variable, t= ^√x_n, we see that

√nµ(Un=m) = 1 2π

Z π√ n

−π√ n

φn

x

√n

e^−ixm/

√ndx

Since,

√1

2πσe^−m2/2nσ2 = 1 2π

Z

R

e^−σ2x2/2e^−ixm/

√ndx