On the central limit theorem for f (n k x)

Transcription

1 Probab. Theory Relat. Fields (1) 146:67 89 DOI 1.17/s On the central limit theorem for f (n k x) Christoph Aistleitner István Berkes Received: 9 May 8 / Revised: 3 September 8 / Published online: 16 December 8 Springer-Verlag 8 Abstract By a classical observation in analysis, lacunary subsequences of the trigonometric system behave like independent random variables: they satisfy the central limit theorem, the law of the iterated logarithm and several related probability limit theorems. For subsequences of the system ( f (nx)) n 1 with π-periodic f L this phenomenon is generally not valid and the asymptotic behavior of ( f (n k x)) k 1 is determined by a complicated interplay between the analytic properties of f (e.g., the behavior of its Fourier coefficients) and the number theoretic properties of n k. By the classical theory, the central limit theorem holds for f (n k x) if n k = k,orif n k+1 /n k α with a transcendental α, but it fails e.g., for n k = k 1. The purpose of our paper is to give a necessary and sufficient condition for f (n k x) to satisfy the central limit theorem. We will also study the critical CLT behavior of f (n k x), i.e., the question what happens when the arithmetic condition of the central limit theorem is weakened infinitesimally. Keywords Lacunary series Central limit theorem Diophantine equations Mathematics Subject Classification () Primary 4A55 6F5 11D4 C. Aistleitner was research supported by FWF grant S963-N13. I. Berkes was research supported by FWF grant S963-N13 and OTKA grants K 615 and K C. Aistleitner (B) Institute of Mathematics A, Graz University of Technology, Steyrergasse 3, 81 Graz, Austria aistleitner@finanz.math.tugraz.at I. Berkes Institute of Statistics, Graz University of Technology, Münzgrabenstrasse 11, 81 Graz, Austria berkes@tugraz.at

2 68 C. Aistleitner, I. Berkes 1 Introduction By classical results of Salem and Zygmund [1,] and Erdős and Gál [8], if a sequence (n k ) k 1 of positive integers satisfies the Hadamard gap condition n k+1 /n k q > 1 (k = 1,,...), (1.1) then (cos πn k x) k 1 satisfies the central limit theorem and the law of the iterated logarithm, i.e., and { lim P x (, 1) : N lim sup N N cos πn k x t } N/ = (t) (1.) N (N log log N) 1/ cos πn k x = 1 a.e. (1.3) where P denotes the Lebesgue measure and is the standard normal distribution function. These results show that lacunary subsequences of the trigonometric system behave like sequences of independent random variables. Actually, much more than (1.) and (1.3) is valid: Philipp and Stout [19] showed that the function S(t) = S(t, x) = cos πn k x (t ) k t considered as a stochastic process over the probability space ([, 1], B, P), isasmall perturbation of a Wiener process {W (t), t } and thus it satisfies several refined asymptotic results for W (t). Typical examples are Chung s law of the iterated logarithm and the arcsin law lim inf (N/ log log k N N) 1/ max cos πn k N l x = π/4 l=1 a.e. lim N P{x (, 1) : A N (x) tn}= π arcsin t where A N (x) denotes the number of positive partial sums k j=1 cos πn j x,1 k N. In view of this remarkable behavior of lacunary trigonometric series, it is natural to ask if a similar result holds for the sequence ( f (n k x)) k 1 for general periodic functions f.aserdős and Fortet (see [16, p. 646]) showed, the answer is negative: if f (x) = cos π x + cos 4π x, then

3 On the central limit theorem for f (n k x) 69 { lim P x (, 1) : N and N f (( k 1)x) t } N = π 1/ t/ cos πs e u du ds N lim sup (N log log N) 1/ f (( k 1)x) = cos π x N a.e. On the other hand, Kac [15] proved that if f : R R is of bounded variation on [, 1] or it is Lipschitz continuous satisfying f (x + 1) = f (x), f (x) dx =, (1.4) then { lim P x (, 1) : N N f ( k x) tσ } N = (t) (1.5) provided σ = f (x)dx + f (x) f ( k x)dx =. (1.6) These results show that the probabilistic behavior of ( f (n k x)) k 1 depends not only on the speed of growth of (n k ) k 1, but also on its number theoretic properties. The sequence ( f (n k x)) k 1 is generally not orthogonal, and the asymptotic evaluation of the integral c k f (n k x) dx (1.7) k N is a difficult problem, closely connected with the behavior of the Dirichlet series a k k s, b k k s

4 7 C. Aistleitner, I. Berkes where f a + (a k cos πkx + b k sin πkx) is the Fourier series of f (see Wintner [6]). The asymptotics of (1.7) is a delicate problem even for lacunary (n k ) k 1, as is shown by the case n k = θ k where θ>1 (not necessarily an integer). Petit [17] and Fukuyama [9] showed that if f satisfies (1.4), f (x) dx = 1 and mild regularity conditions, then f (θ k x) dx c f,θ N (1.8) k N where c f,θ is a constant depending sensitively on f, θ. Ifθ r is irrational for r = 1,,..., then c f,θ = 1, which is exactly the constant we would get if f (θ k x) were independent random variables with mean and variance 1. For other delicate phenomena for k N f (θ k x), see Fukuyama [1]. For a series representation of c f,θ, see Petit [17]. The growth properties of N f (n k x), in particular, estimates for N sup f (n k x) f C for some classes C of functions f satisfying (1.4) play an important role in metric discrepancy theory, see Baker [], Philipp [18] for strong bounds for general, resp. lacunary sequences (n k ). Returning to the CLT, in the lacunary case Gaposhkin [1] showed that holds provided that { lim P x (, 1) : N 1 σn := } N f (n k x) tσ N = (t) (1.9) ( N f (n k x)) dx CN (1.1) for a positive constant C > and (n k ) k 1 satisfies one of the following conditions: (a) n k+1 /n k is an integer for any k 1, (b) lim n k+1/n k = α where α r is irrational for r = 1,,... k (1.11)

5 On the central limit theorem for f (n k x) 71 Takahashi [3] proved that the CLT (1.9) also holds if n k+1 /n k and f Lip (α), α>. The explanation of these phenomena was given in a profound paper of Gaposhkin [13], who showed the following remarkable result: Theorem A Let (n k ) k 1 be an increasing sequence of positive integers satisfying the Hadamard gap condition (1.1) and assume that (1.1) holds for some constant C >. Assume further that for any fixed positive integers a, b, ν the number of solutions of the Diophantine equation an k bn l = ν (k,l 1) (1.1) is bounded by a constant C(a, b), independent of ν. Then the central limit theorem (1.9) holds. Let (n (d) k ) denote the set-theoretic union of the sequences (n k ) k 1,(n k ) k 1,..., (dn k ) k 1. Clearly each element of (n (d) k ) can be represented at most in d different ways in the form jn k,1 j d, k 1 and thus the Diophantine condition in Theorem A is equivalent to saying that for each d 1 the number of solutions of the equation n (d) k n (d) l = ν (k,l 1) is at most C = C(d), uniformly in ν>. The last condition is trivially satisfied if (n (d) k ) satisfies the Hadamard gap condition for each d 1, and it is not hard to see that this is the case if n k+1 /n k and in examples (a), (b) in (1.11) above. Theorem A reveals the Diophantine background of the central limit theorem (1.9), but, as the example at the end of this section will show, its condition is far from necessary for the CLT. The purpose of the present paper is to improve Theorem A and to find the precise condition for the central limit theorem (1.9). Given a sequence (n k ) of positive integers, define for any d 1, ν Z, L(N, d,ν) = #{1 a, b d, 1 k,l N : an k bn l = ν} L(N, d) = sup L(N, d, ν). (1.13) ν> Our main result is the following. Theorem 1.1 Let (n k ) k 1 be a sequence of positive integers satisfying the Hadamard gap condition (1.1) and let f be a function of bounded variation satisfying (1.4) and (1.1). Assume that for any fixed d 1 we have L(N, d) = o(n) as N. (1.14) Then the central limit theorem (1.9) holds. If f is a trigonometric polynomial of order r, it suffices to assume (1.14) for d = r.

6 7 C. Aistleitner, I. Berkes As we will see, the Diophantine condition (1.14) in Theorem 1.1 is best possible: for any d 1, δ> there exists a sequence (n k ) k 1 of positive integers satisfying the Hadamard gap condition such that L(N, d) δn for N N (d,δ)and the CLT (1.9) fails for a trigonometric polynomial of order d. Condition (1.1) is inevitable, as is shown by the example f (x) = cos π x cos 4π x, n k = k, for which the Diophantine condition of Theorem 1.1 is satisfied, but the CLT is not. Of special interest is the following case, where we can calculate the variance σ N explicitly. Slightly modifying the definition of L(N, d) in (1.13), let L (N, d) = sup L(N, d,ν). ν For ν = we exclude the trivial solutions a = b, k = l from L(N, d,ν). Put also f = ( f (x) dx) 1/. Theorem 1. Let (n k ) k 1 be a sequence of positive integers satisfying the Hadamard gap condition (1.1) and let f be a function of bounded variation satisfying (1.4) and f >. Assume that for any fixed d 1 we have L (N, d) = o(n) as N. Then the central limit theorem (1.9) holds with σ N = f N. The difference between the conditions of Theorems 1.1 and 1. is that in Theorem 1. we bound the number of solutions of an k bn l = ν, 1 k,l N (1.15) also for ν =. It is easily seen that Theorem 1. applies if n k+1 /n k, and in example (b) in (1.11) above. In contrast, in case (a) in (1.11), Theorem 1. may fail, as the example n k = k shows. In this case Eq. (1.15) has too many (namely cn) solutions for ν = e.g., if a = 1, b = and in Kac s theorem (1.5) the norming factor is σ N with σ defined by (1.6) and not f N. Finally, it is easy to see that Theorem 1. applies also if n k = k + h(k), k k where h is a nonconstant polynomial with integer coefficients. Theorems 1.1 and 1. show that the validity of the CLT for f (n k x) is determined by the Diophantine functions L(N, d) and L (N, d). Actually, much more than this is valid: even if the the CLT fails, the number of solutions of the Diophantine equation (1.15) is directly connected with the magnitude of the deviation ( sup P t R σ 1 N for large N. To make this precise, let ) N f (n k x) t (t) 1 ε d = lim N L(N, d). N

7 On the central limit theorem for f (n k x) 73 Clearly L(N, d,ν) d N and consequently L(N, d) d N and ε d d. Note that the quantity ε d depends also on the sequence (n k ). Theorem 1.3 Let (n k ) k 1 be a sequence of positive integers satisfying the Hadamard gap condition (1.1) and let f be a trigonometric polynomial of order d with nonnegative coefficients. Let S N = k N f (n kx). Then lim N sup P (S N tσ N ) (t) Bε 1/5 d, (1.16) t where B is a constant depending on f, d and q. On the other hand, for each d 1 there exists a trigonometric polynomial of order d with nonnegative coefficients and a sequence (n k ) k 1 satisfying the Hadamard gap condition (1.1) such that lim N sup P (S N tσ N ) (t) C(d)εd 3 (1.17) t where C(d) is a constant depending only on d. Thus if ε d is small, i.e., (n k ) almost satisfies the conditions of Theorems 1.1 and 1., then f (n k x) almost satisfies the central limit theorem. As the proof of Theorem 1.3 will show, εd 3 in the lower bound in (1.17) can be improved to εd (log ε 1 d ) 1. However, closing the gap between the exponents 1/5 and in the upper and lower bounds seems to be a very difficult problem and we did not pursue it in the present paper. In case of trigonometric polynomials with nonnegative coefficients, Theorem 1.3 quantifies quite precisely the connection between the number of solutions of the Diophantine equation (1.15) and the distribution of S N /σ N for large N. Without the nonnegativity condition the situation is much more complicated: in this case the constant B in Theorem 1.3 should be replaced by B/C, where C is the constant in (1.1), which itself depends on (n k ). To decide for which f and (n k ) condition (1.1) holds is a difficult question not investigated in the present paper. A typical case for an almost CLT of the type (1.16) is when classical numbertheoretic criteria for the CLT are infinitesimally weakened. For example, the CLT holds for f (n k x) if n k+1 /n k α where α r is irrational for r = 1,,...For rational α this criterion fails, but the CLT almost holds if in the reduced form α = p/q both p and q are large. More precisely, for any f of bounded variation satisfying (1.4) and any ε>there exists a K = K (ε, f ) such that if n k+1 /n k α where α = p/q where p and q are coprime integers exceeding K then the left hand side of (1.16)isat most ε. The same phenomenon holds if α r is irrational for 1 r < s and α s = p/q with p, q large. A further example for a near CLT is when n k+1 /n k q, k = 1,,... with q large. Such a result was proved earlier in Berkes [3], see Theorem 4.1 on p. 36. The following example illustrates the difference between our Diophantine condition (1.14) and Gaposhkin s condition in Theorem A. Example Let (m k ) k 1 be a sequence of positive integers with m k+1 m k and let the sequence (n k ) k 1 consist of the numbers m k 1, k = 1,,..., plus the numbers mk+1 1 for the indices k of the form k =[n α ], α>. Let f be a periodic

8 74 C. Aistleitner, I. Berkes Lipschitz function with mean and f = 1. By a result of Takahashi [3], the central limit theorem holds for f (( m k 1)x) with the norming sequence N. Clearly N f (n k x) = M j=1 f (( m j 1)x) + O(N 1/α ) where N N 1/α M N for N N, which implies that f (n k x) also satisfies the CLT. On the other hand, for infinitely many l we have n l = m k 1, n l+1 = m k+1 1forsomek and thus n l+1 n l = 1. The number of such l s up to N is N 1/α and thus the equation n i n j = 1 has at least cn 1/α solutions for the indices 1 i, j N. Consequently, Gaposhkin s number theoretic condition fails for (n k ). In conclusion we note that Gaposhkin s condition implies the validity of the CLT for all subsequences of f (n k x) as well, and for this stronger version of the CLT, Gaposhkin s condition is necessary. However, since different subsequences of f (n k x) can have totally different CLT behavior for arithmetic reasons, it is preferable to give conditions implying the CLT for f (n k x) for a specific sequence (n k ) k 1, without referring to subsequences of (n k ) k 1. In the case when (n k ) k 1 grows subexponentially, i.e., when n k+1 /n k 1, the asymptotic behavior of f (n k x) becomes much more complicated than in the Hadamard lacunary case and the central limit theorem generally fails even for f (x) = cos π x, f (x) = sin π x. A precise condition for the CLT for (cos n k x) k 1 was obtained by Erdős [7], see Takahashi [4,5] for additional information. For the CLT for trigonometric series with small gaps see Berkes [4] and Bobkov and Götze [6] introducing a completely new method in gap theory. For recent asymptotic results for f (n k x) for subexponential (n k ) k 1 see e.g., Philipp [], Fukuyama and Petit [11] and Aistleitner and Berkes [1]. Proof of Theorems 1.1 and 1. In the proof of our theorems, we will use the following theorem by Heyde and Brown [14]: Theorem B Let (Y n, F n, n 1) be a martingale difference sequence with finite fourth moments, let V M = M E(Yi F i 1) and let (b M ) M 1 be any sequence of positive numbers. Then sup t P((Y 1 + +Y M )/ ( M EY 4 b M <t) (t) A i +E ( (V M b M ) ) b M ) 1/5, where A is an absolute constant. In [14] this result is only stated for b M = M EYi, but the proof remains valid for general b M without any change (see [5, Theorem A]). To simplify the formulas we will prove Theorems 1.1 and 1. only in the case when f is an even function; the general case requires only minimal changes. Let f (x) a j cos π jx j=1

9 On the central limit theorem for f (n k x) 75 be the Fourier expansion of f. Without loss of generality we may assume that f 1 and Var f 1, where Var f denotes the total variation of f on the interval [, 1].This implies a j j 1, j 1 (see Zygmund [7, p. 48]). Let ε> be given. We put d = ε 3 +1, p(x) = d a j cos π jx, r(x) = j=1 j=d+1 a j cos π jx. Lemma.1 N f (n k x) C N, N p(n k x) C N, N r(n k x) C ε 3 N Here, and in the sequel, C denotes positive numbers, not always the same, depending only on q, while c denotes positive numbers depending on q and d (and therefore on ε as well). Proof The second inequality is a special case of the first. The other two inequalities follow from N a j cos πn k jx dx 1 1 ( jnk = j n k ) jj j=j+1 1 k k N j =J+1 n k j n k 1 k k N 1 k k N q k k j =J+1 j, j =J+1 1 j { CN for J = Cd 1 N for J = d. Lemma. For any function f satisfying (1.4) we have b a f (λx) dx 1 λ f (x) dx 1 λ f for any real numbers a < b and any λ>. Proof The lemma follows from b a f (λx) dx= 1 λ λb λa f (x) dx= 1 λ λa+k λa f (x) dx+ λb λa+k f (x) dx = 1 λ λb λa+k f (x) dx, where k is the integer with λa + k λb <λa + k + 1.

10 76 C. Aistleitner, I. Berkes We begin with the proof of Theorem 1.1. By the assumptions of the theorem, for any d 1 there exists a function g(n) = g d (N) such that L(N, d,ν) N/g(N) for any ν>. (.1) We divide the set of positive integers into consecutive blocks 1, 1,,,..., i, i,...of lenghts 4log q i and i 1/, respectively. Let i and i + denote the smallest, respectively largest integer in i. Clearly n (i 1) + n i q 4log q i i 4. (.) For every k i 1 i let i = i(k) be defined by k i, put m(k) = log n k + log i and approximate p(n k x) by a discrete function ϕ k (x) such that the following properties are satisfied: v (P1) ϕ k (x) is constant for x < v+1, v =, 1,..., m(k) 1 m(k) m(k) (P) ϕ k (x) p(n k x) ci (P3) E(ϕ k (x) F i 1 ) = where F i denotes the σ -field generated by the intervals [ v m(i+ ), v + 1 ), v =, 1,..., m(i+) 1. m(i+ ) Since p(x) is a trigonometric polynomial, it is Lipschitz-continuous, and thus p(n k x) p(n k x ) cn k m(k) ci for v m(k) x, x < v + 1 m(k) and v< m(k). Thus it is possible to approximate p(n k x) by discrete functions ˆϕ k (x) that satisfy (P1) and (P). For k i and any atom I of the σ -field F i 1 (an interval of length m((i 1)+) ) we get, letting I denote the length of I, 1 I I ˆϕ k (x) dx 1 I I p m((i 1)+ ) n i p(n k x) dx + 1 I + c i 1+log i+log n (i 1) + n i = 4i n (i 1) + n i c i + c i I + c i c i dx

11 On the central limit theorem for f (n k x) 77 by Lemma., (.) and since p f + Var f by (4.1) of Chapter II and (1.5) and (3.5) of Chapter III of Zygmund [7]. Every x [, 1) is contained in an interval of type I for some v, soweputϕ k (x) = ˆϕ k (x) I 1 I ˆϕ k (t) dt for x I and have functions that satisfy (P1), (P) and (P3). We put Y i = k i ϕ k (x), T i = k i p(n k x), T i = k i p(n k x), V M = M E(Y i F i 1). Clearly ϕ k (x), k i are F i measurable and thus Y i is also F i measurable. Let also w i = p(n k x) k i ( M ) 1/ dx and s M = w i. We observe that w i = d a j cos π jn k x dx k i j=1 and get = k,k i 1 j, j d = i p + a j a j k,k i,k >k (cos π(jn k j n k )x) + (cos π(jn k + j n k )x) dx 1 j < j d Ti w i = p(n k x) w i k i = d a j cos π jn k x k i j=1 = 1 j, j d, k, k i < jn k j n k < i n (i 1) j, j d, k, k i i n (i 1) + jn k j n k < n i 1 ( jnk = j n k ) a j a j w i 1 a ja j cos π(jn k j n k )x 1 a ja j cos π(jn k j n k )x + R i (x) = U i (x) + W i (x) + R i (x). (.3)

12 78 C. Aistleitner, I. Berkes Here R i is a sum of at most d i trigonometric functions with coefficients at most 1 and frequencies at least n i. Therefore by Lemma. with f (x) = cos π x, E(R i F i 1 ) 4d i m((i 1)+ ) n i 8d i (i 1) n (i 1) + n i ci 1. (.4) The number of summands in U i and the number of summands in W i (all of them trigonometric functions with coefficients at most 1) are bounded by ci 1/, because the number of quadruples ( j, j, k, k ) with 1 j, j d, k, k i,forwhich< jn k j n k < n i,isatmostd i ( 1 + log q (d + 1) ). In fact, for fixed j, j and k in the case n k >(d+1)n k we have jn k j n k < jn k j (d+1)n k = ( j j (d + 1) ) n k = (d (d + 1)) n k < n k n i and there are at most 1 + log q (d + 1) indices k k for which n k (d + 1)n k (and similarly in case n k < n k /(d + 1)). In particular U i ci 1/ and W i ci 1/. (.5) Clearly Yi Ti p(n k x) ϕ k (x) p(n k x) + ϕ k (x) k i k i ci c ci i ci 1. (.6) k i k i Therefore by (.3) and (.6) wehave V M s M = = By (.4) wehave M M M E(Y i F i 1) s M (.7) E(T i F i 1) s M + c log M E((T i w i ) F i 1 ) + c log M M M M E(U i F i 1 ) + E(W i F i 1 ) + E(R i F i 1 ). M E(R i F i 1 ) c log M. (.8)

13 On the central limit theorem for f (n k x) 79 To estimate M E(W i F i 1 ), we observe ( M E E(W i F i 1 )) E E(W i F i 1 )E(W i F i 1). (.9) 1 i i M By (.5), M E (W i F i 1 ) M ci cm. (.1) For i < i, since E(W i F i 1 ) is F i 1 -measurable, ( ) E E(W i F i 1 )E(W i F i 1) F i 1 = E(Wi F i 1 )E(W i F i 1 ) W i E(W i F i 1 ) ci 1/ E(W i F i 1 ), whence by integration E ( E(W i F i 1 )E(W i F i 1) ) ci 1/ E E(W i F i 1 ). (.11) W i can be written as a trigonometric polynomial of the form n i u=(i ) n (i 1) + c u cos πux, where u c u ci 1/. Thus using Lemma. with f (x) = cos π x we get E(W i F i 1 ) n i u=(i ) n (i 1) + c u u 1 m((i 1)+ ) m((i 1)+) (i ) 1 n (i 1) + ci (i ) 5/ n (i 1) + n (i 1) + n i u=(i ) n (i 1) + c u ci (i ) 5/ q (i 1)+ (i 1) + ci (i ) 5/ q (i 1) 1/. (.1)

14 8 C. Aistleitner, I. Berkes Combining the estimates (.9) (.1), we get M E(W i F i 1 ) cm + 1 i<i M 1/ ci 5/ i 5/ q (i 1) 1/ cm. (.13) Finally, we estimate M E(U i F i 1 ). Note that U i is a sum of trigonometric functions with frequencies at most i n (i 1) +, i.e., U i (x) = i n (i 1) + u=1 c u cos πux, where u c u ci 1/. Hence the fluctuation of U i on any atom of F i 1 is at most i n (i 1) + u=1 i n (i 1) + c u πu m((i 1)+) π i n (i 1) + m((i 1)+ ) ci 1/ n (i 1) + i 1 i n (i 1) + and consequently, which gives = ci 7/ E(U i F i 1 ) U i ci 7/, u=1 c u M M E(U i F i 1 ) U i + c. (.14) The largest frequency of the trigonometric functions in M U i is at most M n (M 1) +, so we can write, grouping the terms with equal frequency, whereby(.1) M U i (x) = M n (M 1) + u=1 d u cos πux, M ( i + i ) d u ( M ) c M3/ + M log M g ( i + i ) g(m) c M3/ g(m) (.15)

15 On the central limit theorem for f (n k x) 81 (without loss of generality we assume that g is nondecreasing) and consequenty u d u M ci 1/ cm 3/. Thus M U i M n (M 1) + u=1 d u c M3/ g(m) c M 3 g(m) M n (M 1) + u=1 d u and hence by (.14) it follows that M E(U i F i 1 ) ( c M 3 ) 1/ + c. (.16) g(m) Substituting the estimates (.8), (.13) and (.16) into(.7), we get V M sm c log M + cm + ( c M 3 ) 1/ + c g(m) and therefore E ((V M sm )) c M3 g(m) + cm5/ c M3 g(m), since we can assume g(x) x 1/. Now we estimate M EYi 4. By Lemma.1 and property (P) we have EY i Ci 1/, and so EY 4 i ( Y i ) EY i Ci 3/ and M EY 4 i CM 5/. Hence by Theorem B we get, using again g(x) x 1/,

16 8 C. Aistleitner, I. Berkes sup P ((Y 1 + +Y M )/s M < t) (t) t ( M EYi 4 + E ( (V M sm A )) s 4 M ) 1/5 ( ) 1/5 ) 1/5 M 5/ + M 3 /g(m) M ca ca( 3 /g(m). (.17) s 4 M s 4 M Now let a positive integer N be given. There exists an M = M(N) with N M CN /3 such that N ( M+1 M+1 ) and therefore N M ( i + i ) M+1 + M+1 CN1/3. We put ˆN = M ( i + i ). Then N M M M f (n k x)= Y i + (T i Y i )+ T i + We put We observe that 1 s M p(n k x) dx = k M i N k= ˆN+1 N σ N = f (n k x). 4 k, k M i (k, k ) M ( i i ) 4 C C p(n k x)+ 1 j, j d N r(n k x). (.18) M p(n k x) p(n k x) dx k i k M i 1 ( jnk = j n k ) jj M k i k i 1 i =1 i 1 j d j=1 1 j M k i M k i k i 1 i =1 i q n k j n k k i 1 i =1 i q M i 1/ i 3/ i 4 C. 4log q i k k

17 On the central limit theorem for f (n k x) 83 Thus by Minkowski s inequality and Lemma.1 N σ N f (n k x) + f (n k x) + r(n k= ˆN+1 k k x) + p(n M i k k x) M i k M i ( M ) 1/ (CN 1/3 ) 1/ +(CN /3 4log q CN /3 ) 1/ + Cε 3 i +(s s M + CN 1/3 log N + C ε 3 N. M +C)1/ A similar calculation yields σ N s M CN 1/3 log N C ε 3 N. Since by assumption σ N KN, choosing ε so small that K C ε 3 >, we get sm (σ N CN 1/3 log N C ε N) 3 CN (.19) and thus 1 C ε 3 σ N /s M 1 + C ε 3. (.) By (.18) we have for any fixed t and <ε<1 ( N ) ( M ) ( ) N P f (n k x) tσ N P Y i (t + ε)σ N + P r(n k x) >εσ N /4 ) ( ) M + P( T M i >εσ N /4 +P (Y i T i ) >εσ N /4 N + P p(n k x) >εσ N /4. (.1) k= ˆN+1 Also, a lower bound for P( N f (n k x) tσ N ) is obtained if in the second line of (.1) we replace t + ε by t ε and change the sign of the four subsequent terms to negative. By property (P) and N ˆN CN 1/3 the last two summands in (.1)are

18 84 C. Aistleitner, I. Berkes zero if N is large enough. By Minkowski s inequality M T M d i = a j cos π jn k x k i j=1 d M a j cos π jn k x j=1 and thus by Chebyshev s inequality ( M P T i k i ( M ) 1/ c i cn 1/3 log N ) >εσ N /4 cn 1/3 (log N). The third relation of Lemma.1 and another application of Chebyshev s inequality yield ( ) N P r(n k x) >εσ N /4 Cε. Therefore by (.17) (.1) wegetfor<ε ε, N N (ε) and any t R, ( N ) P f (n k x) tσ N (t) ( M ) P Y i s M ((t + ε)σ N /s M ) ((t + ε)σ N /s M ) + ((t + ε)σ N /s M ) (t) + Cε + cn 1/3 (log N) ( ) 1/5 ( ca M 3 g(m) 1 sm 4 ) + (1 + Cθ ε 3 )(t + ε) (t) + Cε + cn 1/3 (log N) c(n g( N) 1 N ) 1/5 + Cε + cn 1/3 (log N) cg( N) 1/5 + Cε + ε (.) for some θ with θ 1. Here we used the fact that ((1+Cθ ε 3 )(t +ε)) (t) = O(ε) for any t R with an absolute constant in the O, which can be seen separately, using the mean value theorem, for t and t >, observing that in the case t > and <ε ε,anyξ between t and (1 + Cθ ε 3 )(t + ε) satisfies ξ t / and thus (ξ) e t /8. By the remark after (.1), the difference

19 On the central limit theorem for f (n k x) 85 (t) P( N f (n k x) tσ N ) can be estimated similarly as in (.), except that t +ε in the second, third and fourth line should be replaced by t ε. Since <ε ε was arbitrary, Theorem 1.1 is proved. To prove Theorem 1., assume that L (N, d) = o(n) for any d 1. Then the function g(n) = g d (N) in (.1) can be chosen so that (.1) remains valid also for ν =. Letting ε> and using the same notations as above, we get N N σ N p(n k x) r(n k x) ( N 1/ p(n k x)) dx C ε 3 N N d j=1 a j 1 k<k N 1 j, j d ( f N r N N g(n) f N C ε 3 N 1 ( jnk = j n k ) ) 1/ C ε 3 N for sufficiently large N. A similar argument yields 1/ C ε 3 N σ N f N + C ε 3 N for sufficiently large N and since ε>was arbitrary, it follows that σ N f N. Thus Theorem 1. follows from Theorem Proof of Theorem 1.3 The upper bound in Theorem 1.3 is implicit in the proof of Theorem 1.1. We prove the lower bound in the case d = ; the modifications in the case d 3 are straightforward. Let (m l ) l 1 be a sequence of integers satisfying m l+1 m l.fixr 1 and let (n k ) k 1 consist of the blocks H l ={ m l+1 1, m l+ 1, m l+4 1,..., m l+ r 1 1}, l l, where l is chosen so large that all the above blocks are disjoint. Clearly lim inf k n k+1 /n k = and thus (n k ) k 1 satisfies the Hadamard gap condition. Let a, b N and consider the equation an i bn j = ν (3.1)

20 86 C. Aistleitner, I. Berkes where ν>, i > j. Since m l+1 m l, there exists a constant K = K (a, b) such that if n i and n j (i > j) belong to different blocks H l, H l with l>l, l K, then an i bn j an i /. Hence in this case Eq. (3.1) impliesthatan i / ν an i and thus ν/a n i ν/a. Because of the large separation between the blocks caused by m l+1 m l, the last relation determines uniquely the block H l to which n i belongs and thus the number of choices for n i is at most H l =r. Once n i is known, n j is uniquely determined by (3.1). Thus the number of solutions (i, j) of (3.1) where n i and n j belong to different blocks is at most r + K 1 (a, b), where the second term is due to the number of solutions (i, j) where n i H l, n j H l with 1 l, l < K (a, b). If i and j belong to the same block H l, then (3.1) can be written as i.e., a( m l+ s 1) b( m l+ t 1) = ν s, t r 1 (3.) m l (a s b t ) = ν + a b. (3.3) For c =, the equation a s b t = c has at most one solution (s, t) with s = t. If e.g., s > t and t > log log (b/a), then a s /b t = (a/b) t ( s t 1) (a/b) t > and thus the equation a s b t = c implies c a s < c which is satisfied for at most one s. Since s determines t uniquely, and the number of solutions (s, t) with t log log (b/a) is clearly log log (b/a), it follows that the number of solutions (s, t) of the equation a s b t = c with c =, s > t is at most 1 + log log (b/a). Ifc =, then we have a s = b t, whence log (b/a) = s t = t ( s t 1) t provided e.g., s > t, i.e., the number of such solutions is at most log log (b/a). Summarizing, we have proved that the number of solutions (s, t) of (3.1) such that i and j belong to the same fixed block H l is at most C(a, b), and the number of solutions (s, t) such that i and j belong to different blocks is at most r + K 1 (a, b). Let now N > r and choose M 1sothatMr < N (M +1)r. Then the number of solutions of (3.1)fori, j N (which means that n i and n j are permitted to run in the first M +1 blocks) is at most C(a, b)(m + 1) + r + K 1 (a, b) C(a, b)m + r + K 1 (a, b) C(a, b)n/r +N/r +K 1 (a, b) C 1 (a, b)n/r if N r. Thus L(N, d,ν) C(d)N/r, L(N, d) C(d)N/r for N r and thus ε d = lim sup N 1 L(N, d) C(d)/r. (3.4) N Let now f (x) = cos π x + cos 4π x, then f (n k x) = cos πn k x + cos 4πn k x. k H l k H l k H l The frequencies in the first trigonometric sum on the right side are m l+1 1, m l+ 1, m l+4 1, m l+8 1,... (3.5)

21 On the central limit theorem for f (n k x) 87 and in the second sum are m l+, m l+3, m l+5, m l+9,... (3.6) (The frequency of cos πλx is meant as λ.) Note that the second frequency in (3.5) and the first frequency in (3.6) differ by 1 and thus the sum of the corresponding cosines is cos(π x) cos π( m l+ 3/)x. The remaining frequencies in (3.5) and (3.6) give a trigonometric sum g l (x) with r terms, with frequencies between ml+1 1 and m l+ r Note that, given any 1 < q <, the frequencies in g l satisfy the Hadamard gap condition with ratio q provided l>l (q) and this property remains valid even if we include ml+ 3/ in the above set of frequencies. Thus discarding the first l blocks H l and shifting indices, we can assume that the above statements are valid for all l 1. Thus letting ψ(x) = cos(π x) we have f (n k x) = ψ(x) cos π( ml+ 3/)x + g l (x) k H l and consequently 1 Nr k Nr 1 f (n k x) = ψ(x) Nr N l=1 cos π( m l+ 3/)x + 1 N(r ) cos πp j x =: Z N (x) (3.7) Nr j=1 where p 1 < p <...is Hadamard lacunary, moreover it remains lacunary together with the frequencies m l+1 3/ in the first sum on the right hand side of (3.7). The limit distribution of Z N (x) is easy to determine for any continuous function ψ on [, 1]. Assume first that ψ = s j=1 d j I [v j 1,v j ) is a stepfunction where = v < v 1 <... < v s = 1. Applying the CLT of Salem and Zygmund (see [1], statement (iii) on p. 333) for the interval [v j 1,v j ), it follows that and thus lim N v j ( ( 1 exp(itz N (x))dx = exp 1 v j v j 1 v j 1 lim N exp(itz N (x))dx = 1 1 r + d j r ) ) t exp(h(x)t )dx (3.8)

22 88 C. Aistleitner, I. Berkes where h(x) = 1 (1 ) 1r ψ(x) + r = 1 ψ(x). (3.9) 4r By a simple approximation argument, (3.8) and (3.9) remain valid for any continuous function ψ and thus (Nr) 1/ k Nr f (n kx) has a limit distribution G whose characteristic function is ϕ(t) = exp(h(x)t )dx with h(x) = 1 cos π x 1 r By Taylor expansion we get for t 1 ϕ(t) = e t / 1 t r cos π xdx+ t4 8r ) = e t / (1 + t4 16r + O(r 3 ) = 1 cos π x. r cos π xdx+ O(r 3 ) where the constant in the O is absolute. Thus ϕ(t) e t / B/r for 1/ t 1, r r for some positive absolute constant B. On the other hand, for any two distribution functions F 1 and F with characteristic functions ϕ 1 and ϕ we have for any T ϕ 1 (1) ϕ (1) 3T sup F 1 (x) F (x) + x df 1 (x) + df (x). (3.1) x T x T Using this for F 1 =, F = G, the two integrals on the right side of (3.1) are O(e ct ) and thus choosing T = log r we get for r r that sup (x) G(x) C/(r log r), x which, together with (3.4), completes the proof of Theorem 1.3. Choosing r large, we also proved the remark after Theorem 1.1, concerning the optimality of Theorem 1.1. Acknowledgments The authors are grateful to the referee for several valuable comments and suggestions leading to an improvement of our paper. References 1. Aistleitner, C., Berkes, I.: On the law of the iterated logarithm for the discrepancy of n k x. Monatshefte Math. (to appear). Baker, R.C.: Metric number theory and the large sieve. J. Lond. Math. Soc. 4, 34 4 (1981)

23 On the central limit theorem for f (n k x) Berkes, I.: On the asymptotic behaviour of f (n k x) I-II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34, and (1976) 4. Berkes, I.: A central limit theorem for trigonometric series with small gaps. Z. Wahrsch. Verw. Gebiete 47, (1979) 5. Berkes, I.: On the central limit theorem for lacunary trigonometric series. Anal. Math. 4, (1978) 6. Bobkov, S., Götze, F.: Concentration inequalities and limit theorems for randomized sums. Probab. Theory Relat. Fields 137, (7) 7. Erdős, P.: On trigonometric sums with gaps. Magyar Tud. Akad. Mat. Kut. Int. Közl. 7, 37 4 (196) 8. Erdős, P., Gál, I.S.: On the law of the iterated logarithm. Proc. Kon. Nederl. Akad. Wetensch. 58, (1955) 9. Fukuyama, K.: The central limit theorem for Riesz Raikov sums. Probab. Theory Relat. Fields 1, (1994) 1. Fukuyama, K.: The law of the iterated logarithm for discrepancies of {θ n x}. Acta Math. Hungar. 118, (8) 11. Fukuyama, K., Petit, B.: Le théorème limite central pour les suites de R. C. Baker. Ergodic Theory Dyn. Syst. 1, (1) 1. Gaposhkin, V.F.: Lacunary series and independent functions. Russian Math. Surv. 1, 3 8 (1966) 13. Gaposhkin, V.F.: The central limit theorem for some weakly dependent sequences. Theory Probab. Appl. 15, (197) 14. Heyde, C.C., Brown, B.M.: On the departure from normality of a certain class of martingales. Ann. Math. Stat. 41, (197) 15. Kac, M.: On the distribution of values of sums of the type f ( k t). Ann. Math. 47, (1946) 16. Kac, M.: Probability methods in some problems of analysis and number theory. Bull. Am. Math. Soc. 55, (1949) 17. Petit, B.: Le théorème limite central pour des sommes de Riesz-Raikov. Probab. Theory Relat. Fields 93, (199) 18. Philipp, W.: Limit theorems for lacunary series and uniform distribution mod 1. Acta Arith. 6, ( ) 19. Philipp, W., Stout, W.F.: Almost sure invariance principles for partial sums of weakly dependent random variables. Memoirs of the AMS No. 161 (1975). Philipp, W.: Empirical distribution functions and strong approximation theorems for dependent random variables. A problem of Baker in probabilistic number theory. Trans. Am. Math. Soc. 345, (1994) 1. Salem, R., Zygmund, A.: On lacunary trigonometric series. Proc. Nat. Acad. Sci. USA 33, (1947). Salem, R., Zygmund, A.: La loi du logarithme itéré pour les séries trigonométriques lacunaires. Bull. Sci. Math. 74, 9 4 (195) 3. Takahashi, S.: A gap sequence with gaps bigger than the Hadamards. Tôhoku Math. J. 13, (1961) 4. Takahashi, S.: On lacunary trigonometric series. Proc. Jpn. Acad. 41, (1965) 5. Takahashi, S.: On lacunary trigonometric series II. Proc. Jpn. Acad. 44, (1968) 6. Wintner, A.: Diophantine approximation and Hilbert space. Am. J. Math. 64, (1944) 7. Zygmund, A.: Trigonometric series, 3rd edn, vol. I, II. Cambridge Mathematical Library. Cambridge University Press, London ()