We study zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We prove that the horizontal limiting measure of the zeroes exists almost surely, and that it is non-random if and only if the spectral measure is continuous (or degenerate). In this case, the limiting measure is computed in terms of the spectral measure. We compare the behavior with Gaussian analytic function with symmetry around the real axis. These results extend a work by Norbert Wiener.

Abstract. We study the probability of a real-valued stationary process to be positive on a large interval [0, N ]. We show that if in some neighborhood of the origin the spectral measure of the process has density which is bounded away from zero and infinity, then the decay of this probability is bounded between two exponential functions in N . This generalizes similar bounds obtained for particular cases, such as a recent result by Artezana, Buckley, Marzo, Olsen.

Let f be a zero mean continuous stationary Gaussian process on R whose spectral measure vanishes in a δ-neighborhood of the origin. Then the probability that f stays non-negative on an interval of length L is at most e −cδ 2 L 2 with some absolute c > 0 and the result is sharp without additional assumptions.1991 Mathematics Subject Classification. 60G10, 60G15.

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We show that for any centered stationary Gaussian process of integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in [0, T ] is within ηT of its mean value, up to an exponentially small in T probability. IntroductionThe study of zeroes of Gaussian stationary processes goes back at least to Kac [10] and Rice [19]. Since then, much work on this topic appeared in the statistics, physics, mathematics and engineering literature. One of the earliest and most fundamental results in this area is the Kac-Rice formula, which calculates the mean number of zeroes in any interval. A similar formula may be written for the variance of the number of zeroes, but it is much harder to analyze. It was only many years after Kac and Rice that fluctuations and central limit theorems were better understood, with works by Cuzick [5], Slud [21], Azaïs-León [2] and others. Questions of large deviations, that is, estimation of the rare event of having many more or much less zeroes than expected in a long interval, remained almost unexplored. One particular such event is that of having no zeroes at all in a long interval, which is also known by the name of "persistence". Results (and speculations) about this event were initiated by Slepian [20] and were better understood only recently [8].In the meantime, complex zeroes of certain Gaussian analytic functions received much attention. Most notably, zeroes of the Fock-Bargmann model were introduced by Sodin-Tsirelson [23] and extensively studied by many authors since then. This model has a remarkable property: its zeroes form a point process in the plane with quadratic repulsion, and invariance of distribution under all planar isometries. Sodin-Tsirelson proved the asymptotic normality of these zeroes in [23], and moreover, an exponential concentration of the zeroes around the mean in [24] (See also [12,17] for more about concentration, and [16] for finer results on asymptotic normality). Exponential concentration was proved for other related models, such as nodal lines of spherical harmonics in [15]. Inspired by these works, the question of concentration for real zeroes got some attention [22], [26, Thm. 2c1], but, until now, was not settled even for a single particular example.The aim of this paper is to prove exponential concentration for real zeroes of certain Gaussian stationary functions on R, which have an analytic extension to a strip in the complex plane and smooth spectral density. These conditions allow us to use tools from complex analysis, thus generalizing the mechanics of the aforementioned works on the Fock-Bargmann model.We consider here centered Gaussian stationary processes {X(t) : t ∈ R} having a.s. absolutely continuous sample path. That is, random absolutely continuous functions f : R → R, whose finite