The winding of stationary Gaussian processes

Abstract

This paper studies the winding of a continuously differentiable Gaussian stationary process f : R → C in the interval [0, T ]. We give formulae for the mean and the variance of this random variable. The variance is shown to always grow at least linearly with T , and conditions for it to be asymptotically linear or quadratic are given. Moreover, we show that if the covariance function together with its second derivative are in L 2 (R), then the winding obeys a central limit theorem. These results correspond to similar results for zeroes of real-valued stationary Gaussian functions by Malevich, Cuzick, Slud and others.

An asymptotic formula for the variance of the number of zeroes of a stationary Gaussian process

“…In the setting of complex zeroes of a random Gaussian analytic f : C → C, a linear lower bound for the variance, and an L 2 -condition that guarantees linearity were given in [12]. The same phenomena were then discovered for the winding number of a Gaussian stationary f : R → C in [5].…”

“…In the case of super-linear variance and a slightly stronger mixing condition than (5) we give a precise estimate.…”

“…since this implies that Var[N (T )] = o(T 2 ), by stationarity. Assume first that ρ has no atoms; we adapt the proof of [5,Thm 4]. By the Fomin-Grenander-Maruyama theorem, f is an ergodic process (see, e.g., [15,Sec.…”

Variance linearity for real Gaussian zeros

“…Feldheim [9] has also studied zeros of Gaussian analytic functions (GAFs) whose law is invariant only under horizontal shifts, reduced to a horizontal strip of the form R × [a, b], with −∞ < a < b < ∞. In a related work, Buckley and Feldheim [2] study the winding number of a GAF X : R → C. A strong motivation of their work is the analogy between the winding number and the number of zeros. Some convenient identities coming from complex analysis can provide explicit expressions for these indexes and their moments, whereas the number of zeros of a real Gaussian process is mostly studied in the literature through its Wiener-Ito expansion, which is sometimes not amenable to analysis; one can also use directly Kac-Rice formula [1] but this does not ease the task of estimating the variance.…”

“…Some convenient identities coming from complex analysis can provide explicit expressions for these indexes and their moments, whereas the number of zeros of a real Gaussian process is mostly studied in the literature through its Wiener-Ito expansion, which is sometimes not amenable to analysis; one can also use directly Kac-Rice formula [1] but this does not ease the task of estimating the variance. In particular, we cannot obtain an expression as explicit as (6) in [2] or Section 3.2 in [9] for the number of zeros. It will be interesting to observe in Section 2.1 that the results of these two works about GAFs are similar to the results we obtain here: the variance of the number of zeros (or the winding number) is always at least linear, it is not linear if some square integrability conditions related to the covariance functions are not satisfied, and it is quadratic if the spectral measure has atoms.…”