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We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long–memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener–Itô integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2. [ABSTRACT FROM AUTHOR]

The estimation of the slowness vector of infrasound waves propagating across an array is a critical process leading to the determination of parameters of interest such as the direction of arrival. The sensors of an array are often considered to be located in a horizontal plane. However, due to topography, the altitudes of the sensors are not identical and introduce a bias on the estimate if neglected. However, the unbiased 3D estimation procedure, while suppressing the bias, leads to an increase of the variance. Accounting for an a priori constraint on the slowness vector significantly reduces the variance and could therefore improve the performance of the estimation if the introduced bias by incorrect a priori information remains negligible. This study focuses on measuring the benefits of this approach with a thorough investigation of the bias and variance of the constrained 3D estimator, which is not available in the existing literature. This contribution provides such computations based on an asymptotic Gaussian approximation. Simulations are carried out to assess the theoretical results both with synthetic and real data. Thus, a constrained 3D estimator is proposed yielding the best bias/variance compromise if good knowledge of the propagation wave speed is accessible.