A Central Limit Theorem for Inner Functions
aa r X i v : . [ m a t h . C V ] J un A CENTRAL LIMIT THEOREM FOR INNER FUNCTIONS
ARTUR NICOLAU AND OD´I SOLER I GIBERT
Abstract.
A Central Limit Theorem for linear combinations of iterates of an inner functionis proved. The main technical tool is Aleksandrov Desintegration Theorem for Aleksandrov-Clark measures. Introduction and main results
Inner functions are analytic mappings from the unit disc D into itself whose radial limits areof modulus one at almost every point of the unit circle B D . Inner functions were introducedby R. Nevanlinna and after the pioneering work of brothers Riesz, Frostmann and Beurling,they have become a central notion in Analysis. See for instance [Gar07]. Any inner function f induces a mapping from the unit circle into itself defined at almost every point z P B D by f p z q “ lim r Ñ f p rz q . This boundary mapping will be also called f . It is well known that if f p q “
0, normalized Lebesgue measure m in the unit circle is invariant under this mapping,that is, m p f ´ p E qq “ m p E q for any measurable set E Ă B D . Several authors have also studiedthe distortion of Hausdorff measures by this mapping. See [FP92] and [LNS19]. Dynami-cal properties of the mapping f : B D Ñ B D , as recurrence, ergodicity, mixing, entropy andothers have been studied by Aaranson [Aar78], Crazier [Cra91], Doering and Ma˜ne [DM91],Fern´andez, Meli´an and Pestana [FMP07], [FMP12], Neurwirth [Neu78], Pommerenke [Pom81],and others. Dynamical properties of inner functions have been recently used in several prob-lems on the dynamics of meromorphic functions in simply connected Fatou components. See[BFJK17], [BFJK19] and [EFJS19].It is well known that in many senses lacunary series behave as sums of independent randomvariables. Salem and Zygmund ([SZ47] and [SZ48]) proved a version of the Central LimitTheorem for lacunary series and, a few years later, Weiss proved a version of the Law of theIterated Logarithm in this context ([Wei59]). Our main result is a Central Limit Theorem forlinear combinations of iterates of an inner function fixing the origin. It is worth mentioningthat in our result no lacunarity assumption is needed. Recall that a sequence of measurablefunctions t f N u defined at almost every point in the unit circle converges in distribution to a(circullary symmetric) standard complex normal variable if for any measurable set K Ă C ,one has lim N Ñ8 m pt z P B D : f N p z q P K uq “ π ż K e ´| w | { dA p w q . As it is usual we denote by f n the n -th iterate of the function f . Theorem 1.
Let f be an inner function with f p q “ which is not a rotation. Let t a n u be asequence of complex numbers. Consider σ N “ N ÿ n “ | a n | ` N ÿ k “ f p q k N ´ k ÿ n “ a n a n ` k , N “ , , . . . (1.1) Assume there exists a constant η ą such that lim N Ñ8 sup | a n | : n ď N (´ř Nn “ | a n | ¯ p ´ η q{ “ . (1.2) Both authors are supported in part by the Generalitat de Catalunya (grant 2017 SGR 395) and the SpanishMinisterio de Ci´encia e Innovaci´on (project MTM2017-85666-P).
Then ? σ N N ÿ n “ a n f n converges in distribution to a standard complex normal variable. Let f be an inner function fixing the origin. Then it is well known that Lebesgue measure m is ergodic. Hence the classical Ergodic Theorem gives thatlim N Ñ8 N N ÿ n “ f n p z q “ z P B D . This can be understood as a version of the Law of Large Num-bers. Our result provides the corresponding version of the Central Limit Theorem. Actuallytaking a n “ n “ , . . . , in Theorem 1, one can easily show that lim N Ñ8 σ N { N σ “ σ “ Re 1 ` f p q ´ f p q (1.3)and we deduce the following result. Corollary 2.
Let f be an inner function with f p q “ which is not a rotation. Then ? N N ÿ n “ f n converges in distribution to a complex normal variable with mean and variance σ given by (1.3) , that is, for any measurable set K Ă C , we have lim N Ñ8 m ˜ z P B D : p N q ´ { N ÿ n “ f n p z q P K +¸ “ πσ ż K e ´| w | { σ dA p w q . Observe that when f p q is close to 1 and hence f is close to be the identity map, thevariance σ is large. However if f p q is close to a unimodular constant different from 1, thevariance is small. On the opposite side, if f p q “ σ “ H be the Hardy space of analytic functions in D whose Taylor coefficients are squaresummable. Let t a n u be a sequence of complex numbers. It is easy to show (see Theorem 9)that ř n a n f n converges in H if and only if ř n | a n | ă 8 . A repetition of the proof of ourmain result gives the following version of the Central Limit Theorem for the tails. Theorem 3.
Let f be an inner function with f p q “ which is not a rotation. Let t a n u be asquare summable sequence of complex numbers. Consider σ p N q “ ÿ n ě N | a n | ` ÿ k ě f p q k ÿ n ě N a n a n ` k , N “ , , . . . (1.4) Assume there exists a constant η ą such that lim N Ñ8 sup | a n | : n ě N (`ř n ě N | a n | ˘ p ´ η q{ “ . (1.5) Then ? σ p N q ÿ n “ N a n f n converges in distribution to a standard complex normal variable. Let S N “ ř Nn “ | a n | . It is easy to show (see Theorem 9) that there exists a constant C “ C p f q ą C ´ S N ď σ N ď CS N , N “ , , . . . . When f p q “ CENTRAL LIMIT THEOREM FOR INNER FUNCTIONS 3 σ N “ S N but in general, both quantities do not coincide. However if the following uniformquasiorthogonality condition holdslim N Ñ8 sup k ď N ˇˇˇř N ´ kn “ a n a n ` k ˇˇˇ S N “ , (1.6)then lim N Ñ8 S N { σ N “ ? S N N ÿ n “ a n f n converges in distribution to a standard complex normal variable.We now make some remarks on the assumption and proof of Theorem 1. Condition (1.2)implies that ř n | a n | “ 8 , but one can not expect this last condition to be sufficient inTheorem 1. However note that if t a n u is bounded, both conditions are equivalent. The proofof Theorem 1 uses two relevant properties of the iterates of an inner function fixing the origin.The first one is that the square of the modulus of the partial sums are uncorrelated. Moreconcretely, given a set A of positive integers, consider the corresponding partial sum ξ p A q “ ÿ n P A a n f n . If A X B “ H , we will show in Theorem 6 that ż B D | ξ p A q| | ξ p B q| dm “ ˆż B D | ξ p A q| dm ˙ ˆż B D | ξ p B q| dm ˙ . (1.7)The second property provides an exponential decay of the higher order correlations of theiterates. More concretely, let ε i “ ε i “ ´ i “ , , . . . , k and n ă . . . ă n k bepositive integers satisfying n j ´ n j ´ ě q ě j “ , . . . , k . Denote ε “ p ε , . . . , ε k q and n “p n , . . . , n k q . For a positive integer n , denote by f ´ n the function defined by f ´ n p z q “ f n p z q , z P B D . We will prove in Theorem 13 that there exists a constant C ą
0, independent of theindices, such that ˇˇˇˇˇż B D k ź j “ f ε j n j dm ˇˇˇˇˇ ď C k k ! | f p q| Φ p ε , n q , k “ , , . . . , (1.8)if q is sufficiently large and where Φ is a certain function depending on the choice of indicesthat satisfies Φ p ε , n q ě kq {
4. The main technical tool in the proof of both properties (1.7)and (1.8) is the theory of Aleksandrov-Clark measures and more concretely, the AleksandrovDesintegration Theorem.The paper is organized as follows. In Section 2 we introduce Aleksandrov-Clark measuresand use them to prove property (1.7). In Section 3 we estimate the L and the L norm of ξ p A q . In Section 4 we prove estimate (1.8). The proof of Theorem 1 is given in Section 5.2. Alekandrov-Clark measures and Property (1.7)We start with an elementary auxiliary result which is just a restatement of the invarianceof Lebesgue measure.
Lemma 4.
Let f be an inner function with f p q “ .(a) Let G be an integrable function on B D . Then ż B D G p f p z qq dm p z q “ ż B D G p z q dm p z q (b) Let k ă j be positive integers. Then ż B D f k f j dm “ f p q j ´ k ARTUR NICOLAU AND OD´I SOLER I GIBERT
Proof of Lemma 4.
We can assume that G is the characteristic function of a measurable set E Ă B D . Since m p f ´ p E qq “ m p E q , the identity (a) follows. Using (a) and Cauchy formula,we have ż B D f k f j dm “ ż B D zf j ´ k p z q dm p z q “ f p q j ´ k . (cid:3) Given an analytic mapping from the unit disc into itself and a point α P B D , the function p α ` f q{p α ´ f q has positive real part and hence there exists a positive measure µ α “ µ α p f q in the unit circle and a constant C α P R such that α ` f p w q α ´ f p w q “ ż B D z ` wz ´ w dµ α p z q ` iC α , w P D . (2.1)The measures t µ α : α P B D u are called the Aleksandrov-Clark measures of the function f .Clark introduced them in his paper [Cla72] and many of their deepest properties were foundby Aleksandrov in [Ale86], [Ale87] and [Ale89]. The two surveys [PS06] and [Sak07] as well as[CMR06, Chapter IX] contain their main properties and a wide range of applications. Observethat if f p q “ µ α are probability measures. Moreover, f is inner if and only if µ α is asingular measure for some (all) α P B D . Assume f p q “
0. Computing the first two derivativesin formula (2.1) and evaluating at the origin, we obtain ż B D z dµ α p z q “ f p q α, α P B D , (2.2)and ż B D z dµ α p z q “ f p q α ` f p q α , α P B D . (2.3)Our main technical tool is Aleksandrov Desintegration Theorem which asserts that m “ ż B D µ α dm p α q (2.4)holds true in the sense that ż B D G dm “ ż B D ż B D G p z q dµ α p z q dm p α q , for any integrable function G on the unit circle. Aleksandrov Desintegration Theorem will beused in our next auxiliary result. Lemma 5.
Let f be an inner function with f p q “ . For k “ , , . . . , p , let n k , j k , be positiveintegers such that max t n k , j k u ă min t n k ` , j k ` u , k “ , . . . , p ´ . (2.5) Then ż B D p ź k “ f n k f j k dm “ p ź k “ ż B D f n k f j k dm. (2.6) Proof of Lemma 5.
We argue by induction on p . Assume (2.6) holds for p ´ n ă j . By part (a) of Lemma 4 we have ż B D p ź k “ f n k f j k dm “ ż B D zf j ´ n p z q p ź k “ f n k ´ n p z q f j k ´ n p z q dm p z q . Let t µ α : α P B D u be the Aleksandrov-Clark measures of the inner function f j ´ n . TheAleksandrov Desintegration Theorem (2.4) gives that last integral can be written as ż B D ż B D zα p ź k “ f n k ´ j p α q f j k ´ j p α q dµ α p z q dm p α q . CENTRAL LIMIT THEOREM FOR INNER FUNCTIONS 5
By (2.2) and part (b) of Lemma 4, we have ż B D z dµ α p z q “ f p q j ´ n α “ α ż B D f n f j dm. Hence ż B D p ź k “ f n k f j k dm “ ˆż B D f n f j dm ˙ ż B D p ź k “ f n k ´ j f j k ´ j dm and we can apply the inductive assumption. The invariance property of part (a) of Lemma 4finishes the proof. (cid:3) Our next result is the first important tool in the proof of Theorem 1.
Theorem 6.
Let f be an inner function with f p q “ . Let A k , k “ , , . . . , p , be finitecollections of positive integers such that max t n : n P A k u ă min t n : n P A k ` u , k “ , . . . , p ´ . (2.7) Consider ξ k “ ÿ n P A k a n f n . Then ż B D p ź k “ | ξ k | dm “ p ź k “ ż B D | ξ k | dm. Proof of Theorem 6.
Al almost every point of the unit circle we have | ξ k | “ ÿ n P A k | a n | ` ÿ p a n a j f n f j ` a j a n f j f n q , where the last sum is taken over all indices n, j P A k with j ą n . Hence ś | ξ k | can be writtenas a linear combination of terms of the form ź f n k f j k , where n k , j k P A k . Observe that (2.7) gives the assumption (2.5) in Lemma 5. Now Lemma 5finishes the proof. (cid:3) Norms of Partial Sums
In this Section we will use Aleksandrov-Clark measures to estimate the L and L norms oflinear combinations of iterates of an inner function fixing the origin. The main result of thisSection is Theorem 9. It is worth mentioning that the asymptotic behavior of the Aleksandrov-Clark measures of iterates of an inner function has been studied in [GN15], but we will notuse their results. As before, if n is a positive integer, we will use the notation f ´ n to denotethe function defined by f ´ n p z q “ f n p z q , for almost every z P B D . We start with a technicalauxiliary result which will be used later. Lemma 7.
Let f be an inner function with f p q “ which is not a rotation. Let ε k “ or ε k “ ´ , k “ , , , .(a) Let n k , k “ , , , , be positive integers with max t n , n u ă min t n , n u . Then I “ I p ε n , ´ ε n , n , n q “ ż B D f ε n f ´ ε n f n f n dm “ . (b) Let n ă n ă n be positive integers and II “ II p ε n , ε n , ε n q “ ż B D f ε n p f ε n q f ε n dm. Then there exists a constant C “ C p f q ą independent of the indices n , n , n , such that | II | ď C | f p q| n ´ n . ARTUR NICOLAU AND OD´I SOLER I GIBERT (c) Let n ă n ă n be positive integers and III “ III p ε n , ε n , ε n q “ ż B D p f ε n q f ε n f ε n dm. Then there exists a constant C “ C p f q ą independent of the indices n , n , n , such that | III | ď if n “ n ` and n ď n ` , and | III | ď C | f p q| n ´ n otherwise.(d) Let n ă n ă n ă n be positive integers and IV “ IV p ε n , ε n , ε n , ε n q “ ż B D f ε n f ε n f ε n f ε n dm. Then there exists a constant C “ C p f q ą independent of the indices n , n , n , n , suchthat | IV | ď C | f p q| n ´ n ` n ´ n if n ´ n ą , and | IV | ď C | f p q| n ´ n if n ´ n ď .Moreover | IV | “ | f p q| n ´ n ` n ´ n if ε ε “ ε ε “ ´ .Proof of Lemma 7. Let C denote a positive constant which may depend on the function f butnot on the indices t n i u , whose value may change from line to line.(a) We can assume that n ă n . Part (a) of Lemma 4 gives that I “ ż B D z ε f ´ ε p n ´ n q p z q f n ´ n p z q f n ´ n p z q dm p z q . Let t µ α : α P B D u be the Aleksandrov-Clark measures of f n ´ n . The Aleksandrov Desinte-gration Theorem (2.4) gives I “ ż B D ż B D z ε α ´ ε f n ´ n p α q f n ´ n p α q dµ α p z q dm p α q . By (2.2) ż B D z ε dµ α p z q “ aα ε , α P B D , where | a | “ | f p q| n ´ n . Since f p q “
0, we deduce | I | “ | f p q| n ´ n ˇˇˇˇż B D f n ´ n p α q f n ´ n p α q dm p α q ˇˇˇˇ “ . (b) We can assume ε “
1. Part (a) of Lemma 4 gives that II “ ż B D z p f ε p n ´ n q p z qq f ε p n ´ n q p z q dm p z q . Let t µ α : α P B D u be the Aleksandrov-Clark measures of f n ´ n . The Aleksandrov Desinte-gration Theorem (2.4) gives II “ ż B D ż B D zα ε f ε p n ´ n q p α q dµ α p z q dm p α q . By (2.2) ż B D z dµ α p z q “ f p q n ´ n α, α P B D . Hence II “ f p q n ´ n ż B D α ` ε f ε p n ´ n q p α q dm p α q Since 1 ` ε ď
3, the modulus of last integral is bounded by C | f p q| n ´ n if n ´ n ą ε “
1. Applying part (a) of Lemma 4 and Aleksandrov DesintegrationTheorem as before, we have
III “ ż B D ż B D z α ε f ε p n ´ n q p α q dµ α p z q dm p α q , CENTRAL LIMIT THEOREM FOR INNER FUNCTIONS 7 where t µ α : α P B D u are the Aleksandrov-Clark measures of g “ f n ´ n . Applying (2.3), weobtain III “ g p q ż B D α ` ε f ε p n ´ n q p α q dm p α q ` g p q ż B D α ` ε f ε p n ´ n q p α q dm p α q . Since 2 ` ε ď
3, both integrals are bounded by C | f p q| n ´ n if n ´ n ą
2, and by 1 if n ´ n ď
2. If n ´ n ą
1, we have that | g p q|{ ` | g p q | ď C | f p q| n ´ n . If n ´ n “ | g p q|{ ` | g p q | ď
2. This proves (c).(d) We can assume ε “
1. Arguing as before we have IV “ f p q n ´ n ż B D α ` ε f ε p n ´ n q p α q f ε p n ´ n q p α q dm p α q If ε “ ´
1, we repeat the argument and prove that | IV | ď | f p q| n ´ n ` n ´ n . Moreover if ε “ ´ ε ε “ ´
1, we have | IV | “ | f p q| n ´ n ` n ´ n , as stated in the last part of(d). If ε “
1, let t µ α : α P B D u be the Aleksandrov-Clark measures of g “ f n ´ n . TheAleksandrov Desintegration Theorem (2.4) gives that last integral can be written as ż B D ż B D z α ε f ε p n ´ n q p α q dµ α p z q dm p α q . (3.1)By (2.3) ż B D z dµ α p z q “ g p q α ` g p q α , α P B D . Hence the double integral in (3.1) can be written as g p q ż B D α ` ε f ε p n ´ n q p α q dm p α q ` g p q ż B D α ` ε f ε p n ´ n q p α q dm p α q . Since 2 ` ε ď
3, both integrals are bounded by C | f p q| n ´ n if n ´ n ą
2, and by 1 if n ´ n ď
2. If n ´ n ą
1, we have that | g p q|{ ` | g p q | ď C | f p q| n ´ n . If n ´ n “ | g p q|{ ` | g p q | ď
2. This proves (d). (cid:3)
We will now prove an elementary auxiliary result which will be used several times.
Lemma 8.
Let A be a collection of positive integers and let t a n u be a sequence of complexnumbers. Fix λ P C with | λ | ă . Then ˇˇˇˇˇ ÿ n,k P A ,k ą n a n a k λ k ´ n ˇˇˇˇˇ ď | λ | ´ | λ | ÿ n P A | a n | . Proof of Lemma 8.
Writing j “ k ´ n we have that ÿ n,k P A ,k ą n a n a k λ k ´ n “ ÿ j ą λ j ÿ n,n ` j P A a n a n ` j , where the last sum is taken over all indices n P A such that n ` j P A . It is also understood thatthis sum vanishes if there is no n P A such that n ` j P A . By Cauchy-Schwarz’s inequality, ˇˇˇˇˇ ÿ n,n ` j P A a n a n ` j ˇˇˇˇˇ ď ÿ n P A | a n | . This finishes the proof. (cid:3)
Let H be the Hardy space of analytic functions in the unit disc g p w q “ ř n ě a n w n , w P D ,such that } g } “ sup r ă ż B D | g p rz q| dm p z q “ ÿ n “ | a n | ă 8 . ARTUR NICOLAU AND OD´I SOLER I GIBERT
Any function g P H has a finite radial limit g p z q “ lim r Ñ g p rz q at almost every z P B D and } g } “ ż B D | g p z q| dm p z q . See [Gar07]. For 0 ă p ă 8 let } g } p denote the L p norm on the unit circle of the function g .Next result provides estimates of the L and L norms of linear combinations of iterates of aninner function. It will be applied to finite linear combinations. For t, z P C , let x t, z y “ Re p tz q be the standard scalar product in the plane. Theorem 9.
Let f be an inner function with f p q “ which is not a rotation and let t a n u be a sequence of complex numbers with ř n | a n | ă 8 . Consider ξ “ ÿ n “ a n f n and σ “ ÿ n “ | a n | ` ÿ k “ f p q k ÿ n “ a n a n ` k . (a) We have } ξ } “ σ and C ´ ÿ n “ | a n | ď σ ď C ÿ n “ | a n | , where C “ p ` | f p q|qp ´ | f p q|q ´ .(b) For any t P C we have ż B D x t, ξ y dm “ | t | σ . (c) There exists a constant C “ C p f q ą independent of the sequence t a n u , such that } ξ } ď C } ξ } .Proof of Theorem 9. At almost every point of the unit circle we have | ξ | “ ÿ n “ | a n | ` h, (3.2)where h “ ÿ n,k ě ,k ą n a n a k f k f n . (3.3)Part (b) of Lemma 4 gives } ξ } “ ÿ n “ | a n | ` ÿ n,k ě ,k ą n a n a k f p q k ´ n , which is the identity in (a). Next we prove the estimate in (a). Part (b) of Lemma 4 givesthat ż B D f k f n dm “ b n,k , where b n,k “ f p q n ´ k if n ě k and b n,k “ f p q k ´ n if n ă k . Hence ›››› ÿ n a n f n ›››› “ ÿ n,k a n a k b n,k . Consider the Toeplitz matrix T whose entries are b n,k “ b n ´ k, , n, k “ , , . . . and its symbol s p z q “ ÿ n “´8 b n, z n , z P B D . CENTRAL LIMIT THEOREM FOR INNER FUNCTIONS 9
It is well known that T diagonalizes and its eigenvalues are contained in the interval in the realline whose endpoints are the essential infimum and the essential supremum of s . See [BG00].Since s p z q “ ´ | f p q| | ´ f p q z | , z P B D , the eigenvalues of T are between C ´ and C . This finishes the proof of part (a). Since f p q “
0, the mean value property gives that ż B D ξ dm “ C p f q denote a positive constant only depending on f whose value may change from line to line. The identity (3.2) gives that at almost every pointof the unit circle, we have | ξ | “ ˜ ÿ n “ | a n | ¸ ` h ÿ n “ | a n | ` p Re h q , where h is defined in (3.3). Observe that ż B D h dm “ ÿ n,k ě ,k ą n a n a k f p q k ´ n . Hence Lemma 8 gives that ˇˇˇˇż B D h dm ˇˇˇˇ ď | f p q| ´ | f p q| ÿ n “ | a n | . (3.4)Next we will prove that there exists a constant C “ C p f q ą ż B D | h | dm ď C ˜ ÿ n “ | a n | ¸ . (3.5)Observe that (3.4) and (3.5) give the estimate in (c). Write c n “ a n ÿ k ą n a k f k f n . Using the elementary identity ˇˇˇˇˇ ÿ n “ c n ˇˇˇˇˇ “ ÿ n “ | c n | ` ÿ n “ c n ÿ j ą n c j , we can write ż B D | h | dm “ A ` B, where A “ ÿ n “ | a n | ż B D ˇˇˇˇˇ ÿ k ą n a k f k ˇˇˇˇˇ dm and B “ ÿ n “ a n ÿ k ą n a k ÿ j ą n a j ÿ l ą j a l ż B D f k f n f j f l dm. (3.6)By part (a) we have ż B D ˇˇˇˇˇ ÿ k ą n a k f k ˇˇˇˇˇ dm ď C p f q ÿ k ą n | a k |
20 ARTUR NICOLAU AND OD´I SOLER I GIBERT and we deduce that A ď C p f q `ř n | a n | ˘ . We now estimate B . If n ă k and n ă j ă l , wehave ˇˇˇˇż B D f n f k f j f l dm ˇˇˇˇ “ | f p q| r ´ n `| l ´ s | , where r “ min t k, j u and s “ max t k, j u . This estimate follows from last statement in part (d)of Lemma 7. Part (b) of Lemma 7 gives that ˇˇˇˇż B D f n p f k q f l dm ˇˇˇˇ ď C p f q| f p q| l ´ n , if n ă k ă l . The sum over j ą n in (3.6) will be splited in three terms corresponding to j ą k , j “ k and j ă k . Then | B | ď C p f qp B ` B ` B q where B “ ÿ n ě | a n | ÿ k ą n | a k | ÿ j ą k | a j | ÿ l ą j | a l || f p q| k ´ n ` l ´ j ,B “ ÿ n ě | a n | ÿ k ą n | a k | ÿ l ą k | a l || f p q| l ´ n ,B “ ÿ n ě | a n | ÿ k ą n | a k | ÿ n ă j ă k | a j | ÿ l ą j | a l || f p q| j ´ n `| l ´ k | . Observe that B “ ÿ n ě | a n | ÿ k ą n | a k || f p q| k ´ n ÿ j ą k | a j | ÿ l ą j | a l || f p q| l ´ j . Applying Lemma 8 we deduce that B ď C p f q `ř n ě | a n | ˘ . Similarly B ď ˜ ÿ k ě | a k | ¸ ÿ n ě | a n | ÿ l ą n | a l || f p q| l ´ n , which again by Lemma 8 is bounded by C p f q `ř k ě | a k | ˘ . Finally B “ ÿ n ě | a n | ÿ k ą n | a k | ÿ n ă j ă k | a j || f p q| j ´ n ¨˝ÿ l ą k | a l || f p q| l ´ k ` ÿ j ă l ď k | a l || f p q| k ´ l ˛‚ . Using the trivial estimate ÿ n ă j ă k | a j || f p q| j ´ n ď ÿ j ą n | a j || f p q| j ´ n , we deduce that B ď B ` B where B “ ÿ n ě | a n | ÿ j ą n | a j || f p q| j ´ n ÿ k ą n | a k | ÿ l ą k | a l || f p q| l ´ k and B “ ÿ n ě | a n | ÿ j ą n | a j || f p q| j ´ n ÿ k ą n | a k | ÿ n ă l ď k | a l || f p q| k ´ l . Applying Lemma 8 we have B ď C p f q `ř n ě | a n | ˘ . Writing t “ k ´ l we have ÿ k ą n | a k | ÿ l ă k | a l || f p q| k ´ l ď ÿ t ě | f p q| t ÿ l ě | a l || a l ` t | ď | f p q| ´ | f p q| ÿ n ě | a n | . We deduce that B ď C p f q `ř n ě | a n | ˘ . This finishes the proof. (cid:3) CENTRAL LIMIT THEOREM FOR INNER FUNCTIONS 11 Higher order correlations
Next we will use Aleksandrov-Clark measures to estimate certain integrals which will appearin the proof of Theorem 1. The main result of this Section is Theorem 13. We start givingbounds for the size of the iterates f n and of their derivatives at the origin. Lemma 10.
Let f be an analytic mapping from the unit disc into itself with f p q “ and ă | f p q| ă . Then, there exists an integer d “ d p f q ą such that | f n p w q| ă | f p q| n p ´ | w |q ´ d , w P D , for every n ě . Proof of Lemma 10.
This is a minor modification of [Pom81, Lemma 2]. We include theargument for completeness. Let us denote a “ | f p q| and consider the function ψ p w q “ w a ` w ` aw , w P D , denote its n -th iterate by ψ n and observe that, by Schwarz’s Lemma and induction, we have | f n p w q| ď ψ n p| w |q , w P D , (4.1)for every n ě . Next we use the construction of the K¨onigs function of ψ (see [Sha93, pp. 89–93]). Define for each n ě g n p w q “ a n ψ n p w q , w P D . It is known that t g n u converges uniformly on compact subsets of D to g p w q “ w ` . . . for w P D , satisfying g p ψ p w qq “ ag p w q . Moreover, for 0 ď x ă g n ` p x q “ ψ p ψ n p x qq a n ` “ g n p x q ` a n ´ g n p x q ` a n ` g n p x q ě g n p x q , so that g n p x q ď g p x q for every n ě . Next, since a ą , there exists δ “ δ p f q ą ψ is univalent on t| w | ă δ u and,thus, ψ n and g n are also univalent in this region by Schwarz’s Lemma. By Koebe DistortionTheorem, there exists ε “ ε p f q ą | g p w q| ă | w | ă ε. Now take x “ ε and, for n ě , let x n ` “ ψ ´ p x n q . Observe that Schwarz’s Lemma implies that x n ` ą x n for every n ě . Let d be a positive integer that will be determined later on. We want to show that g p x q ă p ´ x q ´ d , ď x ď x n , (4.2)for every n ě . By the choice of x , it is clear that (4.2) holds for n “ . Assume that (4.2)holds for n and let x ď x ď x n ` . By construction, we have that 0 ă ψ p x q ď x n . Therefore,we get g p x q “ a g p ψ p x qq ă a p ´ ψ p x qq ´ d “ a ˆ ` ax ` x ˙ d p ´ x q ´ d . Since x ě x “ ε, we get the bound g p x q ă a ˆ ` aε ` ε ˙ d p ´ x q ´ d . Hence, using that a “ | f p q| ă , we can choose d “ d p f q large enough and independent of n so that (4.2) holds. Note that , since x n Ñ , one has in fact that (4.2) is valid for 0 ď x ă . Taking (4.1) and applying (4.2), we get | f n p w q| ď a n g n p| w |q ď a n g p| w |q ă a n p ´ | w |q ´ d as we wanted to see. (cid:3) Lemma 11.
Let f be an analytic mapping from the unit disc into itself with f p q “ and a “ | f p q| ă . Let k, l, n be positive integers with l ď n and consider g p w q “ p f n p w qq k for w P D . Then there exists n “ n p f q ą such that for n ě n we have | g l q p q| l ! ď a kn { . Proof of Lemma 11.
Observe first that if a “ , the result holds trivially. Indeed, if f has azero at the origin of multiplicity m ě , then g has a zero of multiplicity km n at the origin.Thus, if m ě l ď n, we have that g l q p q “ . Assume now that a ą . In this case, Lemma 10 asserts that there is a positive integer d “ d p f q for which | f n p w q| ď a n p ´ | w |q ´ d , w P D , for n “ , , . . . . Hence, Cauchy’s estimate gives | g l q p q| l ! ď max t| g p w q| : | w | “ r u r l ď a kn r l p ´ r q kd , ă r ă . Since l ď n we obtain | g l q p q| l ! ď a kn r n p ´ r q kd , ă r ă . (4.3)Fix r such that a { ă r ă
1. Then there exists n “ n p f, r q such that a n { r n p ´ r q d ď a n { p ´ r q d ă , if n ě n . Since k ě a kn { r n p ´ r q kd ď a n { r n p ´ r q d ă . Hence, estimate (4.3) gives | g l q p q| l ! ď a kn { . (cid:3) Let f be an inner function with f p q “ t µ α : α P B D u be its Aleksandrov-Clarkmeasures. Recall that (2.1) gives that for any α P B D , there exists a constant C α P R suchthat α ` f p w q α ´ f p w q “ ż B D z ` wz ´ w dµ α p z q ` iC α , w P D . Expanding both terms in power series, for any positive integer l we have ż B D z l dµ α p z q “ l ÿ k “ α k ż B D f p z q k z l dm p z q , α P B D . Hence for any integer l , the l -th moment of µ α is a trigonometric polynomial in the variable α of degree less or equal than | l | . We will need to estimate the coefficients of this trigonometricpolynomial. Lemma 12.
Let f be an inner function with f p q “ and a “ | f p q| ă . Let l, n be integerswith ď | l | ď n and let t µ α : α P B D u be the Aleksandrov-Clark measures of f n . Then thereexists a constant n “ n p f q ą such that if n ě n , the coefficients of the trigonometricpolynomial ż B D z l dµ α p z q are bounded by a n { for any α P B D . CENTRAL LIMIT THEOREM FOR INNER FUNCTIONS 13
Proof of Lemma 12.
We can assume l ą
0. Then ż B D z l dµ α p z q “ l ÿ k “ α k g l q k p q l ! , α P B D , where g k p w q “ p f n p w qq k , w P D . Lemma 11 gives that | g l q k p q| l ! ď a kn { if n is sufficiently large. Since k ě
1, the proof is completed. (cid:3)
We are now ready to prove the main result of this Section. As before, if n is a positiveinteger, we will use the notation f ´ n to denote the function defined by f ´ n p z q “ f n p z q , foralmost every z P B D . Theorem 13.
Let f be an inner function with f p q “ and a “ | f p q| ă . Let ď k ď q be positive integers. Let ε “ t ε j u kj “ where ε j “ or ε j “ ´ , and let n “ t n j u kj “ where n ă n ă . . . ă n k are positive integers with n j ` ´ n j ą q for any j “ , , . . . , k ´ . Consider I p ε , n q “ ˇˇˇˇˇ ż B D k ź j “ f ε j n j dm ˇˇˇˇˇ Then there exist constants C “ C p f q ą , q “ q p f q ą independent of ε and of n , suchthat if q ě q we have I p ε , n q ď C k k ! a Φ p ε , n q , k “ , , . . . , where Φ p ε , n q “ ř k ´ j “ δ j p n j ` ´ n j q , with δ j P t , { , u for any j “ , . . . , k ´ , and with δ “ and δ k ´ ě { . In addition, for j “ , . . . , k ´ the coefficient δ j “ if and only if δ j ´ “ . Furthermore, if δ j ´ ą , the coefficient δ j depends on ε j ` , . . . , ε k and n j , . . . , n k for j “ , . . . , k ´ . Proof of Theorem 13.
We first prove the following estimate
Claim . We have I p ε , n q ď | f p q| n ´ n max I pt ε , . . . , ε k u , t n ´ n , . . . , n k ´ n uq , ˇˇˇˇˇż B D z k ź i “ f ε i p n i ´ n q p z q dm p z q ˇˇˇˇˇ , ˇˇˇˇˇż B D z ´ k ź i “ f ε i p n i ´ n q p z q dm p z q ˇˇˇˇˇ + To prove Claim 14 we can assume ε “
1. By Lemma 4 and Aleksandrov DesintegrationTheorem we have I p ε , n q “ ˇˇˇˇˇż B D ż B D zα ε k ź i “ f ε i p n i ´ n q p α q dµ α p z q dm p α q ˇˇˇˇˇ , where t µ α : α P B D u are the Aleksandrov-Clark measures of f n ´ n . By (2.2) we have ż B D z dµ α p z q “ f p q n ´ n α, α P B D . Hence if ε “ ´
1, we obtain I p ε , n q “ a n ´ n I pt ε , . . . , ε k u , t n ´ n , . . . , n k ´ n uq and if ε “
1, we obtain I p ε , n q “ a n ´ n ˇˇˇˇˇż B D z k ź i “ f ε i p n i ´ n q p z q dm p z q ˇˇˇˇˇ . This proves Claim 14. We now prove
Claim . For any integers k, l, j with 0 ă | l | ă j and 0 ă j ă k , we have ˇˇˇˇˇż B D z l k ź i “ j f ε i p n i ´ n j ´ q p z q dm p z q ˇˇˇˇˇ ďď ja p n j ´ n j ´ q{ max | n |ď| l |` B D z n k ź i “ j ` f ε i p n i ´ n j q p z q dm p z q ˇˇˇˇˇ+ By Aleksandrov Desintegration Theorem (2.4) we have ż B D z l k ź i “ j f ε i p n i ´ n j ´ q p z q dm p z q “ ż B D ż B D z l α ε j k ź i “ j ` f ε i p n i ´ n j q p α q dµ α p z q dm p α q , where t µ α : α P B D u are the Aleksandrov-Clark measures of f n j ´ n j ´ . Since l ‰
0, accordingto Lemma 12, the moment ż B D z l dµ α p z q , α P B D , is a polynomial in the variable α of degree at most | l | ă j whose coefficients are bounded by a p n j ´ n j ´ q{ . This proves Claim 15.The proof of Theorem 13 proceeds as follows. We first estimate I p ε , n q by the modulusof one of the three integrals in the right hand side of Claim 14 and the factor a n ´ n , thatcorresponds to choosing δ “ . Note that any of these three integrals involve k ´ f. In addition, the integral yielding the maximum in Claim 14 depends onlyon ε , . . . , ε k and on n , . . . , n k . Now if the integral giving the maximum is the first one,we apply Claim 14 again, obtaining the factor a n ´ n and this gives δ “ δ “ . Otherwise we apply Claim 15, obtaining a factor 2 a p n ´ n q{ , which corresponds to choosing δ “ { . Assume that we have applied this procedure to determine the values of δ , . . . , δ j ´ . We continue applying Claim 14 or 15 depending on which integral is yielding the maximumin the previous step, which depends on ε j ` , . . . , ε k and n j , . . . , n k . Observe that when Claim14 is applied, the number of factors of iterates of f is reduced by two units and we obtainthe factor a n j ` ´ n j ` , which corresponds to fixing δ j “ δ j ` “ . When Claim 15 isapplied, we obtain the factor ja p n j ` ´ n j q{ , corresponding to taking δ j “ { , and the numberof factors of iterates of f is reduced by one unit. We continue applying this process at least k { k ´ ż B D z l f ε k p n k ´ n k ´ q p z q dm p z q , | l | ă k ´ ż B D f ε k ´ p n k ´ ´ n k ´ q f ε k p n k ´ n k ´ q dm. Let g “ f n k ´ n k ´ . The modulus of the first integral is | g l q p q|{ l !. Since | l | ă q ă n k ´ n k ´ ,if q is sufficiently large, Lemma 11 gives that last expression is bounded by a p n k ´ n k ´ q{ . Themodulus of the second integral is bounded by a n k ´ n k ´ . This shows that δ k ´ ě { (cid:3) In the proof of Theorem 1 we will split the partial sum into finitely many terms such thatthe sum of the variances of these terms is asymptotically equivalent to the variance of theinitial partial sum. Next auxiliary result provides sufficient conditions for this splitting.
Lemma 16.
Let t a n u be a sequence of complex numbers and λ P C with | λ | ă . Considerthe sequence σ N “ N ÿ n “ | a n | ` N ÿ k “ λ k N ´ k ÿ n “ a n a n ` k , N “ , . . . CENTRAL LIMIT THEOREM FOR INNER FUNCTIONS 15
For N ą , let A j “ A j p N q , j “ , . . . , M “ M p N q , be pairwise disjoint sets of consecutivepositive integers smaller than N . Consider σ p A j q “ ÿ n P A j | a n | ` ÿ k ě λ k ÿ n P A j : n ` k P A j a n a n ` k , j “ , . . . , M. Let A “ Y A j . Assume lim N Ñ8 ř n P A | a n | ř Nn “ | a n | “ and lim j Ñ8 max t| a n | : n P A j u ř n P A j | a n | “ . (4.5) Then lim N Ñ8 ř Mj “ σ p A j q σ N “ . Proof of Lemma 16.
Let B be the set of positive integers smaller or equal to N which are notin the collection A . Then σ N ´ M ÿ j “ σ p A j q “ A ` B ` C, where A “ ÿ n P B | a n | ,B “ N ÿ k “ λ k ÿ B p k q a n a n ` k , where B p k q “ t n P B , n ď N ´ k u and C “ M ÿ j “ ÿ k ě λ k ÿ A p j,k q a n a n ` k , where A p j, k q “ t n P A j : n ď N ´ k, n ` k R A j u . According to part (a) of Theorem 9 wehave σ N ě ´ | λ | ` | λ | N ÿ n “ | a n | . (4.6)Then, Aσ N ď ` | λ | ´ | λ | A ř Nn “ | a n | which by assumption (4.4), tends to 0 as N Ñ 8 . Similarly | B | σ N ď ` | λ | ´ | λ | ř Nk “ | λ | k ř B p k q | a n || a n ` k | ř Nn “ | a n | By Cauchy-Schwarz’s inequality ÿ B p k q | a n || a n ` k | ď ˜ ÿ n P B | a n | ¸ { ˜ N ÿ n “ | a n | ¸ { and we deduce Bσ N ď | λ | ` | λ |p ´ | λ |q `ř n P B | a n | ˘ { ´ř Nn “ | a n | ¯ {
26 ARTUR NICOLAU AND OD´I SOLER I GIBERT which according to (4.4), tends to 0 as N Ñ 8 . We now estimate C . For any k ě M ÿ j “ ÿ A p j,k q | a n || a n ` k | ď ÿ n P A : n ď N ´ k | a n || a n ` k | ď N ÿ n “ | a n | . Hence, applying (4.6), for any positive integer k we have ř k ě k | λ | k ř Mj “ ř n P A p j,k q | a n || a n ` k | σ N ď | λ | k ` | λ |p ´ | λ |q (4.7)Fix ε ą j “ j p ε q ą t| a n | : n P A j u ď ε ÿ n P A j | a n | (4.8)if j ą j . Pick also k such that | λ | k ă ε . Fix k ď k and note that there are at most k indices n P A j such that n ` k R A j . Hence ÿ A p j,k q | a n || a n ` k | ď k | a n j || a n j ` k | , where n j “ n j p k q P A j is the index in A j with n j ` k ď N , where the product | a n || a n ` k | ismaximum. Hence M ÿ j ě j ÿ A p j,k q | a n || a n ` k | ď k M ÿ j ě j | a n j || a n j ` k | ď k ˜ M ÿ j ě j | a n j | ¸ { ˜ M ÿ j “ | a n j ` k | ¸ { . Note that (4.8) gives that ÿ j ě j | a n j | ď ε ÿ j ě j ÿ n P A j | a n | ď ε N ÿ n “ | a n | . Since there are at most k indices n P A j such that n ` k R A j , we also have M ÿ j “ | a n j ` k | ď k N ÿ n “ | a n | . Applying (4.6) again, we deduce ř k ď k | λ | k ř Mj ě j ř A p j,k q | a n || a n ` k | σ N ď ε { ` | λ | ´ | λ | C , (4.9)where C “ ř k ě | λ | k k { . The estimates (4.7) and (4.9) give that | C | σ N ď ř k ď k | λ | k ř j ă j ř A p j,k q | a n || a n ` k | σ N ` ` | λ |p ´ | λ |q ε ` C ` | λ | ´ | λ | ε { . This finishes the proof. (cid:3)
We close this Section with an elementary result which will be used in the proof of Theorem1.
Lemma 17.
Let t f n u , t g n u be two sequences of measurable functions defined at almost everypoint of the unit circle. Assume that there exists a constant C ą such that the followingconditions hold(a) sup n } f n } ď C and lim n Ñ8 ż B D f n dm “ (b) g n p z q ą ´ C for almost every z P B D and lim n Ñ8 } g n } “ . CENTRAL LIMIT THEOREM FOR INNER FUNCTIONS 17
Then lim n Ñ8 ż B D f n e ´ g n dm “ . Proof of Lemma 17.
Cauchy-Schwarz’s inequality gives ˇˇˇˇż B D f n p e ´ g n ´ q dm ˇˇˇˇ ď } f n } } e ´ g n ´ } . Note that there exists a constant M “ M p C q ą | e ´ x ´ | ď M | x | if x ě ´ C .Hence } e ´ g n ´ } ď M } g n } , n “ , , . . . . This finishes the proof. (cid:3) Proof of Theorem 1
Proof of Theorem 1.
Let S N “ N ÿ n “ | a n | , N “ , , . . . Recall that by part (a) of Theorem 9, we have C ´ σ N ď S N ď Cσ N , N “ , , . . . , where C “ p ` | f p q|qp ´ | f p q|q ´ . Pick 0 ă ε ă η , p N “ S ` εN and q N “ S ´ εN . Let C p f q denotea positive constant only depending on f whose value may change from line to line. The proofis organized in several steps.
1. Splitting the Sum.
In this first step, for N large, we will recursively find indices 0 ď M k ă N k ă M k ` ď N , 1 ď k ď Q N , such that if ξ k “ N k ÿ n “ M k ` a n f n , η k “ M k ` ÿ n “ N k ` a n f n , we have lim N Ñ8 Q N q N “ , (5.1) ››››› N ÿ n “ a n f n ´ Q N ÿ k “ p ξ k ` η k q ››››› ď C p f q p N , (5.2) p N ď N k ÿ n “ M k ` | a n | ď p N , q N ď M k ` ÿ n “ N k ` | a n | ď q N , k “ , , . . . , Q N , (5.3)lim N Ñ8 σ N ››››› N ÿ n “ a n f n ´ Q N ÿ k “ ξ k ››››› “ , (5.4) M k ` ´ N k ě q βN , N k ´ M k ě p γN , k “ , , . . . , Q N ´ , (5.5)where β “ p η ´ ε qp ´ ε q ´ and γ “ p η ` ε qp ` ε q ´ .Pick M “ N be the smallest positive integer such that N ÿ n “ | a n | ě p N . The minimality of N gives that N ÿ n “ | a n | ď p N ` | a N | . Now let M be the smallest positive integer such that M ÿ n “ N ` | a n | ě q N . As before, the minimality of M gives that M ÿ n “ N ` | a n | ď q N ` | a M | . We repeat this process until we arrive at an index N k or M k bigger than N . Let Q N be thenumber of times this process is repeated, that is, k “ , , . . . , Q N . Then R N “ N ÿ M QN ` | a n | ď p N . (5.6)Since N ÿ n “ a n f n ´ Q N ÿ k “ p ξ k ` η k q “ N ÿ n “ M QN ` a n f n , the estimate (5.2) follows from part (a) of Theorem 9. By construction we have p N ď N k ÿ n “ M k ` | a n | ď p N ` | a N k | (5.7) q N ď M k ` ÿ n “ N k ` | a n | ď q N ` | a M k ` | , (5.8)for k “ , , . . . , Q N . Fix δ ą
0. Observe that the assumption (1.2) gives that | a N k | `| a M k ` | ă δq N if N is sufficiently large. Taking δ ă N issufficiently large. Moreover the estimates (5.7) give that p p N ` q N q Q N ď S N ´ R N ď p ` δ qp p N ` q N q Q N , (5.9)if N is large enough. Since p N q N “ S N and because of the estimate (5.6), (5.1) follows from(5.9) tending δ to 0. Observe that M k ` ÿ n “ N k ` | a n | ě q N “ S ´ εN . (5.10)By (1.2), if N is sufficiently large, we have that | a n | ď S N ´ η for any n ď N . We deducefrom (5.10) that S N ´ η p M k ` ´ N k q ě q N and M k ` ´ N k ě q βN . A similar argument showsthat N k ´ M K ě p γN . This proves (5.5). We are now going to prove (5.4). Observe that atalmost every point of the unit circle we have ˇˇˇˇˇ Q N ÿ k “ η k ˇˇˇˇˇ “ Q N ÿ k “ | η k | ` Q N ´ ÿ k “ Q N ÿ j ą k η k η j . Since } η k } ď C p f q q N , we have Q N ÿ k “ ż B D | η k | dm ď C p f q q N Q N ď C p f q q N , (5.11)if N is sufficiently large. On the other hand, if j ą k we have ˇˇˇˇż B D η k η j dm ˇˇˇˇ ď ÿ | a r || a t || f p q| t ´ r , where the sum is taken over all indices r, t with N k ă r ď M k ` and N j ă t ď M j ` . Observethat by (5.5), we have t ´ r ě p γN . Writing l “ t ´ r and applying Cauchy-Schwarz’s inequality,we obtain ˇˇˇˇż B D η k η j dm ˇˇˇˇ ď ÿ l ě p γN | f p q| l ÿ | a r || a l ` r | ď C p f q| f p q| p γN S N . CENTRAL LIMIT THEOREM FOR INNER FUNCTIONS 19
Hence Q N ´ ÿ k “ Q N ÿ j ą k ˇˇˇˇż B D η k η j dm ˇˇˇˇ ď C p f q q N S N | f p q| p γN . (5.12)Using (5.11) and (5.12) we obtain that ››››› Q N ÿ k “ η k ››››› ď C p f q q N , if N is sufficiently large. Since σ N ą C p f q S N “ C p f q p N q N , we deduce thatlim N Ñ8 ›››ř Q N k “ η k ››› σ N “ . (5.13)Now (5.2) and (5.13) give (5.4).The main idea in the rest of the proof is that t η k u are irrelevant while due to (5.5), t ξ k u act as independent random variables.
2. Arranging the Fourier Transform.
Applying (5.4), the proof of Theorem 1 reduces toshow that T N “ ? σ N Q N ÿ k “ ξ k , N “ , , . . . converge in distribution to a standard complex normal variable. By the Levi ContinuityTheorem, it is sufficient to show that for any complex number t we have ϕ N p t q “ ż B D e i x t,T N y dm Ñ e ´| t | { , as N Ñ 8 (5.14)Here x t, w y “ Re p tw q is the standard scalar product in the plane. In this second step of theproof we will show thatlim N Ñ8 ϕ N p t q ´ ż B D Q N ź k “ ˆ ` i x t, ξ k y? σ N ˙ exp ˜ ´ x t, ξ k y σ N ¸ dm “ δ ą
0, consider the sets E k “ t z P B D : | ξ k p z q| ą δS N u , k “ , , . . . , Q N . By part (c) ofTheorem 9 we have } ξ k } ď C p f q p N . Txebixeff inequality and (5.1) give Q N ÿ k “ m p E k q ď C p f q p N Q N δ S N ď C p f q δ q N if N is sufficiently large. For µ ą
1, consider the set E “ z P B D : Q N ÿ k “ x t, ξ k p z qy ą µS N + . By part (a) of Theorem 9 we have } ξ k } ď C p f q p N . Txebixeff inequality and (5.1) give m p E q ď C p f q| t | Q N µq N ď C p f q| t | µ , if N is sufficiently large. Hence the set E “ Q N ď k “ E k satisfies m p E q ď C p f q ˆ δ q N ` | t | µ ˙ . Using the elementary identityexp p z q “ p ` z q exp ˆ z ` o p| z |q ˙ , where o p| z | q{| z | Ñ z Ñ
0, we deduceexp p i x t, T N yq “ ˜ Q N ź k “ ˆ ` i x t, ξ k y? σ N ˙ exp ˜ ´ x t, ξ k y σ N ¸¸ exp ˜ Q N ÿ k “ o ˜ x t, ξ k y σ N ¸¸ Fix ε ą
0. Taking δ ą Q N ÿ k “ o ˜ x t, ξ k p z qy σ N ¸ ď C p f q εµ, z P B D z E. Hence ˇˇˇˇˇż B D z E exp p i x t, T N yq dm ´ ż B D z E Q N ź k “ ˆ ` i x t, ξ k y? σ N ˙ exp ˜ ´ x t, ξ k y σ N ¸ dm ˇˇˇˇˇ ďď ´ e C p f q εµ ´ ¯ ż B D z E Q N ź k “ ˜ ` x t, ξ k y σ N ¸ { exp ˜ ´ x t, ξ k y σ N ¸ dm ď e C p f q εµ ´ . Last inequality follows from the elementary estimate p ` x q { e ´ x { ď x ě
0. Hence ˇˇˇˇˇ ϕ N p t q ´ ż B D Q N ź k “ ˆ ` i x t, ξ k y? σ N ˙ exp ´ ˜ x t, ξ k y σ N ¸ dm ˇˇˇˇˇ ď m p E q ` e C p f q εµ ´ , which proves (5.15). Therefore to prove (5.14) it is sufficient to show that for any t P C onehas lim N Ñ8 ż B D Q N ź k “ ˆ ` i x t, ξ k y? σ N ˙ exp ˜ ´ x t, ξ k y σ N ¸ dm “ exp p´| t | { q . This will follow from Lemma 17 applied to the functions f N “ Q N ź k “ ˆ ` i x t, ξ k y? σ N ˙ ,g N “ σ N Q N ÿ k “ x t, ξ k y ´ | t | . According to Lemma 17 it is sufficient to showsup N } f N } ă 8 , (5.16)lim N Ñ8 } g N } “ , (5.17)lim N Ñ8 ż B D f N dm “ . (5.18)
3. Estimating } f N } . Observe that Q N ź k “ ˜ ` x t, ξ k y σ N ¸ “ ` Q N ÿ k “ k σ kN ÿ x t, ξ j y . . . x t, ξ j k y , where the last sum is taken over all collections of indices 1 ď j ă . . . ă j k ď Q N . Since x t, ξ n y ď | t | | ξ n | , Theorem 6 and part (a) of Theorem 9 give that ż B D x t, ξ j y . . . x t, ξ j k y dm ď C p f q k | t | k p kN . CENTRAL LIMIT THEOREM FOR INNER FUNCTIONS 21
Since the total number of distinct collections of indices j , . . . , j k verifying 1 ď j ă . . . ă j k ď Q N is ` Q N k ˘ , we deduce ż B D Q N ź k “ ˜ ` x t, ξ k y σ N ¸ dm ď ` Q N ÿ k “ ˆ Q N k ˙ C p f q k | t | k p kN k σ kN . Since σ N ě C p f q ´ S N “ C p f q ´ p N q N , we deduce ż B D Q N ź k “ ˜ ` x t, ξ k y σ N ¸ dm ď ` Q N ÿ k “ ˆ Q N k ˙ C p f q k | t | k k q kN “ ˆ ` C p f q | t | q N ˙ Q N . Hence (5.1) gives that } f N } ď exp p C p f q | t | { q if N is sufficiently large. This gives (5.16).
4. Estimating } g N } . Consider the set of indices A k “ t n P N : M k ă n ď N k u , k “ , . . . , Q N . Then ξ k “ ÿ n P A k a n f n , k “ , . . . , Q N . (5.19)Let A “ Ť Q N k “ A k . Observe that (5.4) giveslim N Ñ8 ř n P A | a n | S N “ . This is assumption (4.4) of Lemma 16. Assumption (4.5) follows from (1.2). Thus, Lemma16 gives lim N Ñ8 ř Q N k “ } ξ k } σ N “ . (5.20)Denote λ “ t {| t | . We have g N “ | t | σ N Q N ÿ k “ ´ | ξ k | ` λ ξ k ` λ ξ k ´ σ N ¯ . Applying (5.20), the proof of (5.17) reduces to showlim N Ñ8 ››››› σ N Q N ÿ k “ ψ k ››››› “ , where ψ k “ p| ξ k | ´ } ξ k } q ` λ ξ k ` λ ξ k . Now ››››› Q N ÿ k “ ψ k ››››› “ Q N ÿ k “ } ψ k } ` Q N ´ ÿ k “ Q N ÿ j ą k ż B D ψ k ψ j dm. (5.21)Since | ψ k | ď | ξ k | ` } ξ k } , parts (a) and (c) of Theorem 9 give that } ψ k } ď C p f q p N . Hence Q N ÿ k “ } ψ k } ď C p f q p N Q N and we deduce lim N Ñ8 σ N Q N ÿ k “ } ψ k } “ . The second term in (5.21) is splitted as Q N ´ ÿ k “ Q N ÿ j ą k ż B D ψ k ψ j dm “ A ` B ` C ` D, where A “ Q N ´ ÿ k “ Q N ÿ j ą k ż B D ` | ξ k | ´ } ξ k } ˘ ` | ξ j | ´ } ξ j } ˘ dm, B “ Q N ´ ÿ k “ Q N ÿ j ą k ż B D ` | ξ k | ´ } ξ k } ˘ ´ λ ξ j ` λ ξ j ¯ dm,C “ Q N ´ ÿ k “ Q N ÿ j ą k ż B D ´ λ ξ k ` λ ξ k ¯ ` | ξ j | ´ } ξ j } ˘ dm,D “ Q N ´ ÿ k “ Q N ÿ j ą k ż B D ´ λ ξ k ` λ ξ k ¯ ´ λ ξ j ` λ ξ j ¯ dm. By Theorem 6, } ξ k ξ j } “ } ξ k } } ξ j } if k ‰ j and we deduce A “
0. Since the mean of ξ j overthe unit circle vanishes and at almost every point in the unit circle one has | ξ k | “ ÿ n P A k | a n | ` ÿ n P A k ÿ j P A k ,j ą n a n a j f n f j , (5.22)the integrals in B can be written as a linear combination of ż B D f n f j ´ λ ξ j ` λ ξ j ¯ dm, where n , j P A k and hence max t n , j u ă min t n : n P A j u . According to part (a) of Lemma7, ż B D f n f j ξ j dm “ B “
0. Since the mean of ξ k over the unit circle vanishes, we have C “ λ Q N ´ ÿ k “ Q N ÿ j ą k ż B D ξ k | ξ j | dm. For the same reason, using the formula (5.22), we have ż B D ξ k | ξ j | dm “ ż B D ξ k Re h j dm, where h j “ ÿ r,l P A j ,l ą r a r a l f r f l . Using formula (5.19) to expand ξ k , we obtain ż B D ξ k | ξ j | dm “ E ` F, where E “ ÿ n P A k ÿ r,l P A j ,l ą r a n ż B D p f n q ´ a r a l f r f l ` a r a l f r f l ¯ dm,F “ ÿ n,s P A k : s ą n a n a s ÿ r,l P A j ,l ą r ż B D f n f s ´ a r a l f r f l ` a r a l f r f l ¯ dm. By part (c) of Lemma 7 we have ˇˇˇˇż B D p f n q f r f l dm ˇˇˇˇ ` ˇˇˇˇż B D p f n q f l f r dm ˇˇˇˇ ď C p f q| f p q| l ´ n , if n ă r ă l. We deduce that | E | ď C p f q ÿ n P A k | a n | ÿ r,l P A j ,l ą r | a r || a l || f p q| l ´ n . According to (5.5), we have r ´ n ě q βN for any r P A j and any n P A k , j ą k . Now ÿ r,l P A j ,l ą r | a r || a l || f p q| l ´ n ď | f p q| q βN ÿ t ě | f p q| t ÿ r P A j : r ` t P A j | a r || a r ` t | . CENTRAL LIMIT THEOREM FOR INNER FUNCTIONS 23
By Cauchy-Schwarz’s inequality, last sum is bounded by ř r P A j | a r | ď p N . Hence | E | ď C p f q| f p q| q βN p N (5.23)Similarly, part (d) of Lemma 7 gives that ˇˇˇˇż B D f n f s f r f l dm ˇˇˇˇ ` ˇˇˇˇż B D f n f s f r f l dm ˇˇˇˇ ď C p f q| f p q| l ´ n , n ă s ă r ă l ´ , and ˇˇˇˇż B D f n f s f r f l dm ˇˇˇˇ ` ˇˇˇˇż B D f n f s f r f l dm ˇˇˇˇ ď C p f q| f p q| r ´ n , n ă s ă r ă l, r ě l ´ . Using the trivial estimate | a k | ď S N for any k ď N , we deduce that | F | ď C p f q S N ÿ n,s P A k : s ą n ÿ r,l P A j ,l ą r | f p q| l ´ n . As before, l ´ n ě q βN for any r P A j and any n P A k , j ą k . We deduce | F | ď C p f q S N | f p q| q βN { . Now, the exponential decay in (5.23) and (5.16) give thatlim N Ñ8 Cσ N “ . (5.24)The corresponding estimate for D follows from the estimate ˇˇˇˇż B D ξ k ξ j dm ˇˇˇˇ ď C p f q S N | f p q| q βN , k ă j. As before this last estimate follows from (5.5) and from ˇˇˇˇż B D f n f s f l f t dm ˇˇˇˇ ď C p f q| f p q| t ´ n , n ă s ă l ă t ´ , which follows from part (d) of Lemma 7. This finishes the proof of (5.17).
5. Integrating f N . In this last step we will prove (5.18). Observe that at almost every pointin the unit circle we have f N “ ` Q N ÿ k “ i k k { σ kN ÿ x t, ξ i y . . . x t, ξ i k y , where the second sum is taken over all collections of indices 1 ď i ă . . . i k ď Q N . Fix1 ď i ă . . . i k ď Q N . The integral ż B D x t, ξ i y . . . x t, ξ i k y dm “ ´ k ż B D k ź n “ ` tξ i n ` tξ i n ˘ dm is a multiple of a sum of 2 k integrals of the form t r t l ż B D ξ ε i . . . ξ ε k i k dm, where r ` l “ k , ε i “ ε i “ ´ i “ , . . . , k and we denote ξ i ´ p z q “ ξ i p z q , z P B D .Now, each ξ i is a linear combination of iterates of f , ξ j “ ÿ n P A p j q a n f n . Hence ż B D ξ ε i . . . ξ ε k i k dm “ ÿ n P C k ź j “ a ε j n j ż B D f n ε . . . f n k ε k dm, where ř n P C means the sum over all possible k -tuples n “ t n j u kj “ of indices such that n j P A p i j q for j “ , . . . , k. Since | a n | ď S N , n ď N , we have ˇˇˇˇż B D ξ ε i . . . ξ ε k i k dm ˇˇˇˇ ď S kN ÿ n P C ˇˇˇˇż B D f n ε . . . f n k ε k dm ˇˇˇˇ . Let ε “ t ε j u k ´ j “ be fixed and consider Φ p n q “ Φ p ε , n q “ ř k ´ j “ δ j p n j ` ´ n j q where δ j Pt , { , u for j “ , . . . , k ´ , with δ “ δ k ´ ě { , and with δ j “ δ j ´ “ j “ , . . . , k ´ , as defined in Theorem 13. Let a “ | f p q| . Theorem 13 gives ˇˇˇˇż B D ξ ε i . . . ξ ε k i k dm ˇˇˇˇ ď k ! S kN C p f q k ÿ n P C a Φ p n q . We split the sum over n P C as follows. Let D denote the set of p k ´ q -tuples δ “ t δ j u k ´ j “ of coefficients that can appear in Φ p n q . That is, those tuples with δ j P t , { , u for j “ , . . . , k ´ , with δ “ δ k ´ ě { , and with δ j “ δ j ´ “ , for j “ , . . . , k ´ . Observe that there are less than 2 k such tuples. Given a k -tuple n P C , letus denote by δ p n q the p k ´ q -tuple δ of coefficients appearing in Φ p n q . Then we have that ÿ n P C a Φ p n q “ ÿ δ P D ÿ t n P C : δ p n q“ δ u a Φ p n q . Given δ “ t δ j u k ´ j “ P D , we define Φ δ p n q “ ř k ´ j “ δ j p n j ` ´ n j q for every n P C . We clearlyhave that ÿ n P C a Φ p n q ď ÿ δ P D ÿ n P C a Φ δ p n q . (5.25)Consider now a fixed δ “ p δ , . . . , δ k ´ q , and recall that δ “ . Let l p q be the minimuminteger such that δ l p q` “ l p q “ k ´ δ j ‰ ď j ď k ´ l p q ą , we have that δ j “ { ď j ď l p q by Theorem 13. Thus, to finda bound for the right-hand side of (5.25), we need to estimate sums of the form l ÿ j “ ÿ n j P A p i j q a p n ´ n q`p n l ´ n q{ “ l ÿ j “ ÿ n j P A p i j q a p n ´ n q{ `p n l ´ n q{ (5.26)for some 1 ă l ď k ´ . Denote here n “ max A p i q , n “ min A p i q and n l “ min A p i l q , andobserve that n ´ n ě q βN because of (5.5). Assume l ą
2. Summing over n and n we getthat (5.26) is bounded by Ca q βN { l ÿ j “ ÿ n j P A p i j q a p n l ´ n q{ . Next, summing over n j for j up to l ´ | A p i q| ` . . . ` | A p i l ´ q| , whilesumming over n l we get the factor a p n l ´ n q{ . Here, | A p i j q| denotes the number of indices inthe set A p i j q . Using (5.5), we have that | A p i j q| ě p γN ą q βN for any j “ , . . . , k and, thus, weget that n l ´ n ě q βN ` | A p i q| ` . . . ` | A p i l ´ q| ą lq βN . Hence, we find that l ÿ j “ ÿ n j P A p i j q a p n ´ n q`p n l ´ n q{ ď Ca lq βN { . (5.27)Note that if l “ l “
2, then (5.27) is obvious. Assume now that we have determined l p m ´ q . If l p m ´ q ă k ´ , then let l p m q be the minimum integer such that l p m ´ q ă l p m q ď k ´ δ l p m q` “ . We iterate this process until we set l p r q “ k ´ ď r ď k. Observe that, by Theorem 13, we have that l p m q ě l p m ´ q ` . Taking l p q “ , the full sum over n P C in the right-hand side of (5.25) becomes a product EFERENCES 25 of sums of the form (5.26) with j ranging from l p m ´ q ` l p m q , for m “ , . . . , r. Thus,applying the bound (5.27) we get that ÿ n P C a Φ δ p n q ď r ź m “ Ca p l p m q´ l p m ´ qq q βN { ď C k a kq βN { . Now, summing over δ P D and using the fact that there are at most 2 k such tuples, we getthat ÿ n P C a Φ p n q ď C k a kq βN { . Thus ˇˇˇˇż B D ξ ε i . . . ξ ε k i k dm ˇˇˇˇ ď k ! S kN C p f q k a kq βN { . We deduce that ˇˇˇˇż B D x t, ξ i y . . . x t, ξ i k y dm ˇˇˇˇ ď k ! S kN C p f q k | t | k a kq βN { . Since the total number of collections of indices 1 ď i ă . . . ă i k ď Q N is ` Q N k ˘ , we deducethat ˇˇˇˇż B D f N dm ´ ˇˇˇˇ ď Q N ÿ k “ ˆ Q N k ˙ k !2 ´ k { σ ´ kN p C p f q S N | t |q k a kq βN { . Last sum is smaller than ˜ ` C p f q| t | S N Q N a q βN { ? σ N ¸ Q N ´ , which tends to 0 as N Ñ 8 becauselim N Ñ8 S N Q N a q βN { σ N “ . (cid:3) References [Aar78] J. Aaronson. “Ergodic theory for inner functions of the upper half plane”.
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Universitat Aut`onoma de Barcelona, Departament de Matem`atiques, 08193 Barcelona
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