Naomi Feldheim

Zeroes of Gaussian analytic functions with translation-invariant distribution

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 functions with symmetry around the real axis. These results extend a work by Norbert Wiener.

A note on the convex infimum convolution inequality

We characterize the symmetric measures which satisfy the one dimensional convex infimum convolution inequality of Maurey. For these measures the tensorization argument yields the two level Talagrands concentration inequalities for their products and convex sets in

Long Gaps Between Sign-Changes of Gaussian Stationary Processes

\mathbb{R}^n

Variance of the number of zeroes of shift-invariant Gaussian analytic functions

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The Two-Dimensional Small Ball Inequality and Binary Nets

We study the probability of a real-valued stationary process to be positive on a large interval

Persistence of Gaussian stationary processes: a spectral perspective

[0,N]

Convergence of the Quantile Admission Process with Veto Power.

. 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

On the probability that a stationary Gaussian process with spectral gap remains non-negative on a long interval

N

The winding of stationary Gaussian processes

. This generalizes similar bounds obtained for particular cases, such as a recent result by Artezana, Buckley, Marzo, Olsen.

Exponential concentration for zeroes of stationary Gaussian processes

Following Wiener, we consider the zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We show that the variance of the number of zeroes in a long horizontal rectangle [−T,T] × [a, b] is asymptotically between cT and CT2, with positive constants c and C. We also supply with conditions (in terms of the spectral measure) under which the variance grows asymptotically linearly with T, as a quadratic function of T, or has intermediate growth. The results are compared with known results for real stationary Gaussian processes and other models.