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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]

We consider linear processes, not necessarily Gaussian, with long, short or negative memory. The memory parameter is estimated semi-parametrically using wavelets from a sample X1,..., X n of the process. We treat both the log-regression wavelet estimator and the wavelet Whittle estimator. We show that these estimators are asymptotically normal as the sample size n → ∞ and we obtain an explicit expression for the limit variance. These results are derived from a general result on the asymptotic normality of the scalogram for linear processes, conveniently centred and normalized. The scalogram is an array of quadratic forms of the observed sample, computed from the wavelet coefficients of this sample. In contrast to quadratic forms computed on the basis of Fourier coefficients such as the periodogram, the scalogram involves correlations which do not vanish as the sample size n → ∞. [ABSTRACT FROM AUTHOR]

A quantum key distribution (QKD) network is an infrastructure that allows the realization of the key distribution cryptographic primitive over long distances and at high rates with information-theoretic security. In this work, we consider QKD networks based on trusted repeaters from a topology viewpoint, and present a set of analytical models that can be used to optimize the spatial distribution of QKD devices and nodes in specific network configurations in order to guarantee a certain level of service to network users, at a minimum cost. We give details on new methods and original results regarding such cost minimization arguments applied to QKD networks. These results are likely to become of high importance when the deployment of QKD networks will be addressed by future quantum telecommunication operators. They will therefore have a strong impact on the design and requirements of the next generation of QKD devices. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating the period of an unknown periodic function observed in additive Gaussian noise sampled at irregularly spaced time instants in a semiparametric setting. To solve this problem, we propose a novel estimator based on the cumulated Lomb–Scargle periodogram. We prove that this estimator is consistent, asymptotically Gaussian and we provide an explicit expression of the asymptotic variance. Some Monte Carlo experiments are then presented to support our claims. [ABSTRACT FROM AUTHOR]

We consider a Gaussian time series, stationary or not, with long memory exponent d ∈ ℝ. The generalized spectral density function of the time series is characterized by d and by a function f*(λ) which specifies the short-range dependence structure. Our setting is semi-parametric in that both d and f* are unknown, and only the smoothness of f* around λ = 0 matters. The parameter d is the one of interest. It is estimated by regression using the wavelet coefficients of the time series, which are dependent when d ≠ 0. We establish a Central Limit Theorem (CLT) for the resulting estimator $\hat{d}$. We show that the deviation $\hat{d}-d$, adequately normalized, is asymptotically normal and specify the asymptotic variance. [ABSTRACT FROM AUTHOR]

In recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semi-parametric asymptotic theory, comparable with the one developed for Fourier methods, is still lacking. In this article, we adapt to the wavelet setting, the classical semi-parametric framework introduced by Robinson and his co-authors for estimating the memory parameter of a (possibly) non-stationary process. Our results apply to a class of wavelets with bounded supports, which include but are not limited to Daubechies wavelets. We derive an explicit expression of the spectral density of the wavelet coefficients and show that it can be approximated, at large scales, by the spectral density of the continuous-time wavelet coefficients of fractional Brownian motion. We derive an explicit bound for the difference between the spectral densities. As an application, we obtain minimax upper bounds for the log-scale regression estimator of the memory parameter for a Gaussian process and we derive an explicit expression of its asymptotic variance. [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.

A fast and efficient estimation method is proposed that compensates the distortion in nonlinear transformation models. A likelihood-based estimator is developed that can be computed by an EM-type algorithm. The consistency of the estimator is shown and its limit distribution is provided. The new estimator is particularly well suited for fluorescence lifetime measurements, where only the shortest arrival time of a random number of emitted fluorescence photons can be detected and where arrival times are often modeled by a mixture of exponential distributions. The method is evaluated on real and synthetic data. Compared to currently used methods in fluorescence, the new estimator should allow a reduction of the acquisition time of an order of magnitude.