Zeros of smooth stationary Gaussian processes

Abstract

Let f : R → R be a stationary centered Gaussian process. For any R > 0, let νR denote the counting measure of {x ∈ R | f (Rx) = 0}. Under suitable assumptions on the regularity of f and the decay of its correlation function at infinity, we derive the asymptotics as R → +∞ of the central moments of the linear statistics of νR. In particular, we derive an asymptotics of order R p 2 for the p-th central moment of the number of zeros of f in [0, R]. As an application, we prove a functional Law of Large Numbers and a functional Central Limit Theorem for the random measures νR. More precisely, after a proper rescaling, νR converges almost surely towards the Lebesgue measure in weak- * sense. Moreover, the fluctuation of νR around its mean converges in distribution towards the standard Gaussian White Noise. The proof of our moments estimates relies on a careful study of the k-point function of the zero point process of f , for any k 2. Our analysis yields two results of independent interest. First, we derive an equivalent of this k-point function near any point of the large diagonal in R k , thus quantifying the short-range repulsion between zeros of f . Second, we prove a clustering property which quantifies the long-range decorrelation between zeros of f .

Cumulants asymptotics for the zeros counting measure of real Gaussian processes

“…Theorem 1.6 implies a strong law of large number and a central limit theorem for the sequence of random measure (ν n ) n∈N . The two following Corollaries 1.7 and 1.8 extend the results of [1,Sec. 1.4] to our framework, and we refer to this paper for a more thorough discussion.…”

“…The assumption H 2 (q) is much weaker than the one present in [1], where the authors require a function g that decrease like x −8p . Here we show that the asymptotics of higher moments is independent of the rate of decay of the covariance function, and must only satisfy some uniform square integrability condition.…”

“…As mentioned above, the CLT which is obtained from the above moments asymptotics by the method of moments is already known in the particular case of stationary Gaussian processes with covariance function belonging to the Schwartz class, see [1]. Here the assumption on the decay of the correlation function is greatly relaxed and we only need to assume the square integrability of the covariance kernel as well as its derivatives.…”

“…Observe that the function ρ p,n is ill defined when two of its arguments collapse. This issue is solved by using the technique of divided differences, that appeared in [11] and was developed in [1]. Let us give an example with p = 2.…”

“…Up to now, very few results about the asymptotics of the higher central moments are known. The best result so far is the one by T. Letendre and M. Ancona [1], where it has been proved that the p-th central moment, when properly rescaled, converges towards the p-th moment of a Gaussian random variable, under the very restrictive condition that the covariance function is in the Schwartz class of fast decreasing functions. This last result then yields another proof of the CLT for the number of zeros of this particular class of processes by the method of moments.…”