Sum of normally distributed random variables

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For given random variables (X, Y), functional representation lemma states that there exists a random variable Z and a deterministic function g such that Y = g(X, Z), and X is independent of Z. In other words, Z captures all the information about Y not contained in X. We refer to Z as a ‘perfect functional representation’ if it satisfies I(Y ;Z) = H(Y |X). We show that the existence of perfect representation depends solely on the conditional distribution pY|X and define the corresponding conditional distribu

Due to their numerous applications, in particular in Mechanism Design, Prophet Inequalities have experienced a surge of interest. They describe competitive ratios for basic stopping time problems where random variables get revealed sequentially. A key drawback in the classical setting is the assumption of full distributional knowledge of the involved random variables, which is often unrealistic. A natural way to address this is via sample-based approaches, where only a limited number of samples from the dis

We present a theoretical and numerical analysis of Monte Carlo methods for the estimation of statistical moments of random variables

X:Ω\rightarrow E

taking values in a Banach space

E

. For practical computation, we consider finite-dimensional approximation subspaces

{(E_\ell)_{\ell\in\mathbb{N}}\subset E}

of increasing dimension. We develop a refined error analysis that explicitly accounts for a dependence of the Rademacher type constants on the dimension of

E_\ell

, leading to novel complexity resul

The stochastic gradient descent (SGD) method is a widely used approach for solving stochastic optimization problems, but its convergence is typically slow. Existing variance reduction techniques, such as SAGA, improve convergence by leveraging stored gradient information; however, they are restricted to settings where the objective functional is a finite sum, and their performance degrades when the number of terms in the sum is large. In this work, we propose a novel approach which is well suited when the o

The evolution and improvement of wind LiDAR technologies in the last years have made them more affordable and very powerful tools, and as a consequence they are being extensively used in both the industrial and academic communities but their limitations and the uncertainty of their measurements are not yet completely understood. Wind LiDARs are based in the Doppler effect of backscattered laser light from aerosol and its most important limitation is that it only provides measurements of the wind speed compo

We present a new information-theoretic result which we call the Chaining Lemma. It considers a so-called "chain" of random variables, defined by a source distribution X-(0) with high min-entropy and a number (say, t in total) of arbitrary functions (T-1,.. ...

Factor graph is a representative graphical model to handle uncertainty of random variables. Factor graph has been used in various application domains such as named entity recognition, social network analysis, and credibility evaluation. In this paper, we study the problem of reducing uncertainty in factor graph towards reaching a common truth or deterministic information. We propose a pay-as-you-go approach that leverages user feedback for uncertainty reduction. As the availability of human input is often l

In this paper, a finite element error analysis is performed on a class of linear and nonlinear elliptic problems with small uncertain input. Using a perturbation approach, the exact (random) solution is expanded up to a certain order with respect to a parameter that controls the amount of randomness in the input and discretized by finite elements. We start by studying a diffusion (linear) model problem with a random coefficient characterized via a finite number of random variables. A priori and a posteriori

In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by N random variables. The elliptic problem is remapped on to a corresponding PDE with a fixed deterministic domain. We show that the solution can be analytically extended to a well defined region in CN with respect to the random variables. A sparse grid stochastic collocation method is then used to compute

We study the single-phase flow in a saturated, bounded heterogeneous porous medium. We model the permeability as a log-normal random field. We perform a perturbation analysis, expanding the solution in Taylor series. The series is directly computable in the case of a random field parametrized by a finite number of random variables. On the other hand, in the case of an infinite dimensional random field, suitable equations satisfied by the derivatives of the stochastic solution can be derived. We give a theor

In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the prob

We consider the problem of estimating multiple filters from convolutive mixtures of several unknown sources. We propose to exploit both the time-frequency (TF) sparsity of the sources and the sparsity of the mixing filters. Our framework consists of: a) a clustering step to group the TF points where only one source is active, for each source; b) a convex optimisation step, to estimate the filters using TF cross-relations that capture linear constraints satisfied by the unknown filters. Experiments demonstra

We report on recent results on the spectral statistics of the discrete Anderson model in the localized phase. Our results show, in particular, that, for the discrete Anderson Hamiltonian with smoothly distributed random potential at sufficiently large coupling, the limit of the level spacing distribution is that of i.i.d. random variables distributed according to the density of states of the random Hamiltonian. This text is a contribution to the proceedings of the conference "Spectra of Random Operators and

Thanks to their different senses, human observers acquire multiple information coming from their environment. Complex cross-modal interactions occur during this perceptual process. This article proposes a framework to analyze and model these interactions t ...

In this thesis, we address the problem of face modelling by using dedicated statistical generative models, with an application to the face authentication task. Face authentication consists in either accepting or rejecting a user's claim supported by its fa ...

T lymphocytes (T cells) are key components of the adaptive immune system. These cells are able to recognize an enormous variety of pathogens thanks to the great specificity of their trans-membrane proteins, the T cell receptors (TCRs). TCR diversity is cre ...

A random variable dominates another random variable with respect to the covariance order if the covariance of any two monotone increas- ing functions of this variable is smaller. We characterize completely the covariance order, give strong suﬃcient conditions for it, present a num- ber of examples in concrete economic applications, and provide natural extensions for the multivariate context. In analogy to mean preserving spreads in standard stochastic dominance, we show that the covariance order is intimate

Claude Elwood Shannon in 1948, then of the Bell Labs, published one of the ground breaking papers in the history of engineering [1]. This paper (”A Mathematical Theory of Communication”, Bell System Tech. Journal, Vol. 27, July and October 1948, pp. 379 - 423 and pp. 623 - 656) laid the groundwork of an entirely new scientific discipline, information Theory, that enabled engineers for the first time to deal quantitatively with the elusive concept of information”. In his celebrated work, Shannon cleanly laid

The kurtosis of a signal is a quantitative measure of how

peaky' it is. In this paper we consider two scenarios of communication over fading channels with kurtosis constraints: in the first, we analyze a non-coherent Rayleigh fading channel where the input signal is required to satisfy a kurtosis constraint in addition to a power constraint. In the second, we find the

worst' fading process that satisfies a kurtosis constraint and has a given second moment, while the fading coefficients are assumed to be k

Mixture models form the essential basis of data clustering within a statistical framework. Here, the estimation of the parameters of a mixture of Gaussian densities is considered. In this particular context, it is well known that the maximum likelihood approach is statistically ill posed, i.e. the likelihood function is not bounded above, because of singularities at the boundary of the parameter domain. We show that such a degeneracy can be avoided by penalizing the likelihood function using a suited type o

Sum of normally distributed random variables

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98 Virtual Wind LiDAR Experiments using LES data: Examples of the Limitations of LiDAR Profilers and Scanners

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