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We consider the problem of estimating the mixing density f from n i.i.d. observations distributed according to a mixture density with unknown mixing distribution. In contrast to finite mixtures models, here the distribution of the hidden variable is not restricted to a finite set but is spread out over a given interval.We propose an approach to construct an orthogonal series estimator of the mixing density f involving Legendre polynomials. The construction of the orthonormal sequence varies from one mixture model to another. Minimax upper and lower bounds of the mean integrated squared error are provided which apply in various contexts. In the specific case of exponential mixtures, it is shown that the estimator is adaptive over a collection of specific smoothness classes, more precisely, there exists a constant A > 0 such that, when the order m of the projection estimator verifies m ~ A log( n), the estimator achieves the minimax rate over this collection. Other cases are investigated such as Gamma shape mixtures and scale mixtures of compactly supported densities including Betamixtures. Finally, a consistent estimator of the support of the mixing density f is provided. [ABSTRACT FROM AUTHOR]

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]