Strong approximation of lacunary series with random gaps

We investigate the asymptotic behavior of sums ∑k=1Nf(nkx)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{k=1}^N f(n_kx)$$\end{document}, where f is a mean zero, smooth periodic function on R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document} and (nk)k≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_k)_{k\ge 1}$$\end{document} is a random sequence such that the gaps nk+1-nk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k+1}-n_k$$\end{document} are i.i.d. Our result shows that, in contrast to the classical Salem–Zygmund theory, the almost sure behavior of lacunary series with random gaps can be described very precisely without any assumption on the size of the gaps.


Introduction
Let f : R → R be a measurable function satisfying It is well known that for rapidly increasing (n k ) k≥1 the sequence ( f (n k x)) k≥1 behaves like a sequence of independent random variables over the probability space ([0, 1], B, λ), where B is the Borel σ -algebra on [0, 1] and λ is the Lebesgue measure. For example, if n k+1 /n k → ∞ (1.2) and f satisfies the Lip α condition for some constants and α > 0 and K > 0, then with respect to ([0, 1], B, λ) (see Takahashi [24,25]). Here, and in the sequel, · denotes the L 2 norm. Assuming only the Hadamard gap condition n k+1 /n k ≥ q > 1, k = 1, 2, . . . (1.5) the situation becomes more complicated. Kac [12] proved that f (n k x) satisfies the CLT for n k = 2 k and Erdős and Fortet (see [13], p. 646) showed that this generally fails for n k = 2 k − 1. Gaposhkin [10] showed that f (n k x) satisfies the CLT provided the ratios n k+1 /n k are integers or n k+1 /n k → α > 1 where α r is irrational for r = 1, 2, . . .. A necessary and sufficient number-theoretic condition for the CLT for f (n k x) under (1.5) was given by Aistleitner and Berkes [4]. For a related sufficient criterion for the law of the iterated logarithm for the discrepancy of {n k x} for almost all x, see Aistleitner [1]. For subexponentially growing sequences (n k ), the asymptotic behavior of S N = N k=1 f (n k x) becomes much more complicated and the arising number theoretical problems become essentially intractable. As a consequence, the limit distribution (if exists) of normed sums of f (n k x) is not known even for f (x) = sin 2π x and simple sequences like n k = k r (r = 3, 4, . . .). (In the case of n k = k 2 the limit distribution was found using deep methods, see Jurkat and Van Horne [11], Marklof [14].) In such situations, it is natural to investigate the random case, i.e. when (n k ) is an increasing random sequence, and prove asymptotic results valid for almost all (n k ); in other words, to describe the "typical" behavior of sums N k=1 f (n k x). The simplest model for sequences with random gaps is when the gaps n k+1 −n k are i.i.d. random variables, and in a series of papers Schatte [19][20][21] gave a general study of this model. In particular, Schatte gave metric upper bounds for the discrepancy of {n k x} in a large class of discrete and continuous cases. Schatte's results have been extended and improved by Weber [26], Berkes and Weber [6], Berkes and Raseta [5]; on the other hand, Raseta [18] proved a functional law of the iterated logarithm for sums N k=1 f (n k x) for smooth periodic f . The purpose of the present paper is to prove that in the case of gaps n k+1 − n k with absolutely continuous distribution, the partial sums N k=1 f (n k x) can be closely approximated by a Wiener process, a result having far reaching asymptotic consequences for the sequence f (n k x). More precisely, we will prove the following result.
Theorem 1 Let (X n ) n≥1 be a sequence of i.i.d. random variables defined on a probability space ( , F, P) and let S n = n k=1 X k . Assume X 1 is bounded with bounded density. Let f be a Lip (α) function satisfying (1.1) and put where U is a uniform (0, 1) random variable, independent of (X n ) n≥1 . Then for any fixed x > 0 the series (1.6) is absolutely convergent with P-probability 1, A x, f ≥ 0 and the sequence (X k ) k≥1 can be redefined, without changing its distribution, on a new probability space together with a Wiener process for any ε > 0.
Clearly, the redefinition of (X k ) in Theorem 1 does not change the asymptotic properties of the sums N k=1 f (S k x) and thus limit theorems implied by the approximation (1.7) for the redefined sequence f (S k x) hold for the original sequence defined on ( , F, P) as well.
We note that in Theorem 1 we do not assume X 1 > 0, and thus the sequence (S k ) k≥1 need not be increasing. If E X 1 = 0, then by standard results of probability theory the sequence (S k ) k≥1 is dense in R; otherwise the random walk (S k ) k≥1 is transient and S k tends to +∞ or −∞ almost linearly. The a.s. absolute convergence of the series in (1.6) will follow from the arguments in Sect. 4.
An immediate consequence of Theorem 1 is for almost all x ∈ R and for almost all sequences (S k ) k≥1 generated by the random walk model. The functional versions of these results can also be written out and proved without any problem. For further asymptotic consequences of an approximation result of type (1.7) we refer to Strassen [22] and Philipp and Stout [17].
In view of (1.8) and (1.9), the properties of the function A x, f are of considerable interest and we will investigate them in Sect. 4.
Note that all of the previous consequences of Theorem 1 were almost sure limit theorems and using Fubini's theorem we cannot prove, e.g., that P-a.s. the normed partial . We now formulate a version of Theorem 1 implying a CLT and many related weak limit theorems.
Theorem 2 Under the conditions of Theorem 1 the sequence (X k ) k≥1 can be redefined, without changing its distribution, on a new probability space together with a Wiener process W such that for any ε > 0, where ξ is a random variable uniformly distributed over (0, 1), independent of (X k ) and W .
In other words, we can get an approximation of N k=1 f (n k x) with a single Wiener process W when not only the sequence (n k ) k≥1 , but also the x is randomized. Theorem 2 implies, for example, that where the right hand side denotes the distribution of A 1/2 ξ ζ , where ζ is an N (0, 1) variable independent of ξ . Clearly, this distribution is mixed normal. However, this is a central limit theorem on the square × [0, 1], and whether n −1/2 n k=1 f (S k x) has a mixed Gaussian limit P-a.s. over ([0, 1], B, λ) remains open.
As we see, the limsup resp. liminf in (1.8), (1.9) are functions of x, in contrast to constant limsup and liminf in the case of sums of independent random variables.
Similarly, the limit distribution of normed partial sums in Theorem 2 is a mixed normal distribution, in contrast to standard Gaussian limit in classical situations. In the case of sums N k=1 sin n k x with n k = 2 k − 1, this phenomenon was discovered by Erdős and Fortet (see [13], p. 646); for more general series see Morgenthaler [15], Weiss [27], Gaposhkin [10]. The deeper fact that the limsup in the law of the iterated logarithm for the discrepancy of lacunary sequences {n k x} can also be nonconstant, was proved by Aistleitner [2,3] and Fukuyama [8,9]. See also Berkes and Raseta [5] for the exact value of the limsup in case of the discrepancy of {n k x} for random n k .

Some lemmas
In the Introduction we discussed implications of our theorems for the partial sums N k=1 f (S k x) as a sequence of random variables over different probability spaces. For the rest of the paper, x > 0 will be fixed and we consider f (S k x) as a sequence of random variables over ( , F, P), and the symbols P, E will be meant with respect to this probability space.
Lemma 1 below, which is a slight generalization of Lemma 2 of [18], establishes the near independence of separated block sums of the variables f (S k x). The proof of the present form requires only routine changes.

are independent.
Put m k = k j=1 j 1/2 , m k = k j=1 j 1/4 and let m k = m k + m k . Using Lemma 1 we can construct sequences ( k ) k≥0 , ( k ) k≥0 of random variables such that 0 = 0, and are sequences of independent, mean zero random variables.
Lemma 2 implies A 1 = A 1, f ≥ 0 and similarly we have A x, f ≥ 0 for all x > 0. The series expansion (1.6) resembles the series expansion of the long range variance of a stationary process. The weaker relations were proved in [18], Lemma 2. The proof of the present form uses the same argument with minor changes.

Proof of Lemma 3
With EX 2 1 replaced by an unspecified constant C depending on the distribution of X 1 , this lemma follows from statement (c) of Theorem 1 of Schatte [19]. To get C = EX 2 1 we note that letting p k denote the density of S k and f (r ) = E(e 2πir X 1 ), we have by a formula in the proof of Theorem 1 in Schatte [19], p. 277 and Parseval's relation Relation (2.3) implies that there exists a random variable U * on ( , F, P), uniformly distributed over (0, 1), such that |S k − U * | ≤ ρ n−2 EX 2 1 and thus by the (2.4) The following lemma is a special case of Strassen's strong approximation theorem [23], Theorem 4.4.

Lemma 4
Let Y 1 , Y 2 , . . . be independent r.v.'s with mean 0 and finite fourth moments, let a n = n i=1 EY 2 i and assume ∞ n=1 EY 4 n /a 2ϑ n < ∞ with 0 < ϑ < 1. Then the sequence Y 1 , Y 2 , . . . can be redefined on a new probability space together with a Wiener process W such that Y 1 + · · · + Y n = W (a n ) + o a n (1+ϑ)/4 log a n a.s.

Proof of the theorems
We begin with the proof of Theorem 1. In what follows, C 1 , C 2 , . . . denote positive constants, depending (at most) on the distribution of X 1 . Since together with (X k ) the sequence (X k x) also satisfies the conditions of Theorem 1 for any x > 0, it suffices to prove the theorem for x = 1. We will apply Lemma 4 for the sequences (T k ) k≥1 and (T k * ) k≥1 defined before. Clearly, (T k ) k≥1 is a sequence of independent, mean zero random variables and In [6], pp. 59-60 it is shown that for arbitrary real coefficients (c k ) we have  for some Wiener process W . Define a sequence ( p(n)) n≥1 of integers by Clearly, p(n) ∼ C 4 n 2/3 and, as we have shown above, for some other Brownian motion W . Now .
We estimate each term separately. Since m p(n) ∼ n, we have by (3.2) In view of (2.1), (2.4) and the Lipschitz property of f we have Summarizing the above estimates, we obtain our result.
Proof of Theorem 2 By the theorem of Ionescu Tulcea (see e.g. [16], p. 154), on a suitable probability space one can define jointly a sequence {X * k , k ≥ 1} of r.v.'s, a Wiener process W * and a r.v. ξ uniformly distributed over (0, 1) such that the conditional distribution of the vector ({X * k , k ≥ 1}, W * ) ∈ R ∞ ×C(0, ∞) given ξ = x equals the distribution of the vector ({X k , k ≥ 1}, W (x) ) in Theorem 1. In particular, the conditional distribution of {X * k , k ≥ 1} given ξ = x equals the distribution of {X * k , k ≥ 1} which does not depend on x and thus ξ is independent of {X * k , k ≥ 1}. For the same reason, ξ is independent of W * . Further, by the construction and relation (1.7) of Theorem 1, we have the analogue of (1.10) where S k is replaced by the partial sums S * k = k j=1 X * j and W (n) is replaced by W * (n). This completes the proof of Theorem 2.

Properties of A x, f
In view of (1.8), (1.9), the function A x, f in (1.6) plays an important role in the asymptotic study of N k=1 f (n k x). In this section we study the properties of A x, f . First we give an explicit formula for A x, f in the case f (x) = sin 2π x. Let X * 1 = X 1 −μ, where μ = EX 1 . Since EX * 1 = 0 and since all moments of X * 1 exist by the boundedness of X 1 , the Taylor expansion of the characteristic function ϕ of X * 1 around 0 is where the even order terms give the real part and the odd order terms give the imaginary part. Grouping the even and odd terms, we get where B(x) = b 0 + b 2 x 2 + · · · and C(x) = c 0 + c 2 x 2 + · · · , here b 0 = −2π 2 σ 2 , where σ 2 is the variance of X 1 and c 0 = − 4π 2 3 E(X 1 − μ) 3 .

Lemma 5 We have
As a consequence, A x, f is infinitely many times differentiable for x > 0 and Then for f (x) = sin 2π x we get, using the independence of U and S k , In the Re{. . .} in the last line we have a geometric progression with quotient q = ϕ(2π x)e 2πiμx . Since X * 1 has a density, |q| = |ϕ(2π x)| < 1 for all x > 0 and thus ∞ k=0 q k is finite and we get Since the term for k = 0 of the sum in (4.3) is equal to E f 2 (U ) = 1/2, we have (4.4) Substituting (4.1) and e 2πiμx = cos 2πμx + i sin 2πμx into (4.4) we get, after some algebra, that the Re{. . .} in the second line of (4.4) equals K /L, where K and L are defined above. Clearly, for μ = 0 the Taylor series of K and L start with the term (2π 2 μ 2 − b 0 )x 2 , resp. 4π 2 μ 2 x 2 , and thus the limit of Re{· · · } in the second line of (4.4) as x → 0 is Thus, in view of (4.4) we get the first line of (4.2). For μ = 0 the expansion of L starts with a term later than x 2 and we get the second line of (4.2). Since X 1 has a density, |ϕ(2π x)| < 1 for x > 0 and the boundedness of X 1 implies that all moments of X 1 are finite. Thus the characteristic function ϕ is infinitely many times differentiable, and consequently the right hand side of (4.3) and thus also A x, f are infinitely many times differentiable on (0, +∞). Finally, we study the properties of A x, f for general smooth f .

Lemma 6
Assume that X 1 has a bounded density with a bounded derivative. Then the function A x, f is a continuous function of x for all x > 0 and lim x→∞ A x, f = f 2 .
Proof Applying Lemma 3 for the random variable X 1 = X 1 x it follows that where again, S k x is meant mod 1 and Using the assumptions on the density p of X 1 and integration by parts, we see that and thus the right hand side of (4.5) cannot exceed for any 0 ≤ u ≤ 1 and x ≥ A where C 12 and ρ < 1 are positive constants also depending on A and the distribution of X 1 . Multiplying the last inequality with f (u) and integrating with respect to u we get where U is a uniform (0, 1) random variable independent of V and S k . Since i.e. every term of the sum in (1.6) tends to 0 as x → ∞. Since the series converges uniformly over [A, ∞) for any A > 0, this implies that its sum also converges to 0 as x → ∞, i.e. lim x→∞ A x, f = f 2 . As before, (4.9) will follow if we show that E f (u + S k x) → 0 as x → ∞ for any fixed k ≥ 1 and any u ∈ (0, 1). Since, together with the function f (x), the function f (x + u) also satisfies (1.1), it remains to show that for any fixed k ≥ 1 we have E f (S k x) → 0, or equivalently where g k is the density of S k . To this end we first note that if h(u) is the indicator function of a subinterval (a, b) of (0, 1), then  1 0 | f (v)|dv. It follows then that (4.11) holds for any stepfunction in (0, 1) and thus by a simple approximation argument, for any bounded measurable function in (0, 1). This proves (4.10) and thus (4.9) is established, completing the proof of Lemma 6.