The uniform central limit theorem for theta sums

Jurkat, W. B.; Horne, Jeremy

doi:10.1215/s0012-7094-83-05030-5

Cited by 21 publications

(16 citation statements)

References 8 publications

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“…In particular, the tail asymptotics would now be of order R −4 , and stationarity and rotation-invariance of the process fail. In the special case c 1 = α = 0, a limiting theorem for the absolute value |X N (1)| = N −1/2 |S N (x)| was previously obtained by Jurkat and van Horne [25], [26], [24] with tail asymptotics 4 log 2 π 2 R −4 (see also [9, Example 75]), while the distribution for the complex random variable X N (1) = N −1/2 S N (x) for was found by Marklof [32]; the existence of finitedimensional distribution of of the process t → S tN (x) was proven by Cellarosi [10], [9]. Demirci-Akarsu and Marklof [13], [12] have established analogous limit laws for incomplete Gauss sums, and Kowalski and Sawin [29] limit laws and invariance principles for incomplete Kloosterman sums and Birch sums.…”

Section: 7)mentioning

confidence: 78%

Quadratic Weyl sums, automorphic functions and invariance principles

Cellarosi

Marklof

2016

Proc. London Math. Soc.

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Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation here is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion, non-differentiability), although time increments are not independent and the value distribution at each fixed time is distinctly different from a normal distribution.

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Section: 7)mentioning

confidence: 78%

Quadratic Weyl sums, automorphic functions and invariance principles

Cellarosi

Marklof

2016

Proc. London Math. Soc.

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“…Let us remark that Marklof's approach uses the equidistribution of long, closed horocycles in the unit tangent bundle of a suitably constructed non-compact hyperbolic manifold of finite volume. Moreover, the explicit asymptotics for the moments of N − 1 2 |S a (N )| (along with central limit theorems [12,13,14]) were found by Jurkat and van Horne and generalized by Marklof [17] in the case of more general theta sums using Eisenstein series. In particular it is known that the above distribution function Ψ is not Gaussian.…”

Section: Introductionmentioning

confidence: 82%

“…The proof of (15) follows from (14) and can be recovered mutatis mutandis from the proof of the analogous result for Euclidean continued fractions. See, e.g., [23].…”

Section: Continued Fractions With Even Partial Quotientsmentioning

confidence: 94%

Limiting curlicue measures for theta sums

Cellarosi¹

2011

Ann. Inst. H. Poincaré Probab. Statist.

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We consider the ensemble of curves {γ α,N : α ∈ (0, 1], N ∈ N} obtained by linearly interpolating the values of the normalized theta sum N − 1 2 N −1 n=0 exp(πin 2 α), 0 ≤ N < N . We prove the existence of limiting finite-dimensional distributions for such curves as N → ∞, with respect to an absolutely continuous probability measure µ R on (0, 1]. Our Main Theorem generalizes a result by Marklof [17] and Jurkat and van Horne [12,13]. Our proof relies on the analysis of the geometric structure of such curves, which exhibit spiral-like patterns (curlicues) at different scales. We exploit a renormalization procedure constructed by means of the continued fraction expansion of α with even partial quotients and a renewal-type limit theorem for the denominators of such continued fraction expansions.

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“…Cellarosi [1] has studied the analogous setting for theta sums S N (x) = ∑ [tN ] h=1 e(xh 2 ) with x uniformly distributed with respect to Lebesgue measure, generalizing the limit theorems for theta sums investigated by Marklof [10] and earlier by Jurkat and van Horne [6,7,8]. Cellarosi's proof relies on a renormalization procedure established by means of continued fraction expansion of x and renewal-type limit theorem for the denominators of continued fraction expansion of x.…”

Section: Xq(t)mentioning

confidence: 99%