scholarly journals A CLT FOR INFORMATION-THEORETIC STATISTICS OF NON-CENTERED GRAM RANDOM MATRICES

2012 ◽  
Vol 01 (02) ◽  
pp. 1150010 ◽  
Author(s):  
WALID HACHEM ◽  
MALIKA KHAROUF ◽  
JAMAL NAJIM ◽  
JACK W. SILVERSTEIN
Keyword(s):  
Random Matrices ◽  
Radio Channel ◽  
Random Variable ◽  
Multiple Antenna ◽  
Information Theoretic ◽  
Variable Formula ◽  
Main Motivation ◽  
Unit Variance ◽  
The Moment ◽  
Variance Profile

In this article, we study the fluctuations of the random variable: [Formula: see text] where [Formula: see text], as the dimensions of the matrices go to infinity at the same pace. Matrices Xn and An are respectively random and deterministic N × n matrices; matrices Dn and [Formula: see text] are deterministic and diagonal, with respective dimensions N × N and n × n; matrix Xn = (Xij) has centered, independent and identically distributed entries with unit variance, either real or complex. We prove that when centered and properly rescaled, the random variable [Formula: see text] satisfies a Central Limit Theorem and has a Gaussian limit. The variance of [Formula: see text] depends on the moment [Formula: see text] of the variables Xij and also on its fourth cumulant [Formula: see text]. The main motivation comes from the field of wireless communications, where [Formula: see text] represents the mutual information of a multiple antenna radio channel. This article closely follows the companion article "A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile", Ann. Appl. Probab. (2008) by Hachem et al., however the study of the fluctuations associated to non-centered large random matrices raises specific issues, which are addressed here.

scholarly journals Linear eigenvalue statistics of random matrices with a variance profile

2020 ◽  
pp. 2250004
Author(s):  
Kartick Adhikari ◽  
Indrajit Jana ◽  
Koushik Saha
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Random Variable ◽  
Second Order ◽  
Linear Eigenvalue Statistics ◽  
Eigenvalue Statistics ◽  
Eigenvalue Statistic ◽  
Variation Distance ◽  
Random Matrix Ensembles ◽  
Variance Profile

We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centered, of a random matrix with a variance profile and the standard Gaussian random variable. The second-order Poincaré inequality-type result introduced in [S. Chatterjee, Fluctuations of eigenvalues and second order poincaré inequalities, Prob. Theory Rel. Fields 143(1) (2009) 1–40.] is used to establish the bound. Using this bound, we prove central limit theorem for linear eigenvalue statistics of random matrices with different kind of variance profiles. We re-establish some existing results on fluctuations of linear eigenvalue statistics of some well-known random matrix ensembles by choosing appropriate variance profiles.

scholarly journals Some Useful Integral Representations for Information-Theoretic Analyses

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 707 ◽  
Author(s):  
Neri Merhav ◽  
Igal Sason
Keyword(s):  
Integral Representation ◽  
Random Variables ◽  
Direct Calculation ◽  
Random Variable ◽  
Two Dimensions ◽  
Integral Representations ◽  
Independent Random Variables ◽  
Nonnegative Random Variable ◽  
Information Theoretic ◽  
The Moment

This work is an extension of our earlier article, where a well-known integral representation of the logarithmic function was explored and was accompanied with demonstrations of its usefulness in obtaining compact, easily-calculable, exact formulas for quantities that involve expectations of the logarithm of a positive random variable. Here, in the same spirit, we derive an exact integral representation (in one or two dimensions) of the moment of a nonnegative random variable, or the sum of such independent random variables, where the moment order is a general positive non-integer real (also known as fractional moments). The proposed formula is applied to a variety of examples with an information-theoretic motivation, and it is shown how it facilitates their numerical evaluations. In particular, when applied to the calculation of a moment of the sum of a large number, n, of nonnegative random variables, it is clear that integration over one or two dimensions, as suggested by our proposed integral representation, is significantly easier than the alternative of integrating over n dimensions, as needed in the direct calculation of the desired moment.

scholarly journals A CLT for information-theoretic statistics of Gram random matrices with a given variance profile

2008 ◽  
Vol 18 (6) ◽  
pp. 2071-2130 ◽  
Author(s):  
Walid Hachem ◽  
Philippe Loubaton ◽  
Jamal Najim
Keyword(s):  
Random Matrices ◽  
Information Theoretic ◽  
Variance Profile

Markovian Analysis of Aging Distributed Systems Remaining Life

2014 ◽  
Author(s):  
Anna Bushinskaya ◽  
Sviatoslav Timashev
Keyword(s):  
Distributed Systems ◽  
Markov Process ◽  
Residual Life ◽  
Limit State ◽  
Real Life ◽  
Random Variable ◽  
Operating Pressure ◽  
Remaining Life ◽  
Failure Pressure ◽  
The Moment

Correct assessment of the remaining life of distributed systems such as pipeline systems (PS) with defects plays a crucial role in solving the problem of their integrity. Authors propose a methodology which allows estimating the random residual time (remaining life) of transition of a PS from its current state to a critical or limit state, based on available information on the sizes of the set of growing defects found during an in line inspection (ILI), followed by verification or direct assessment. PS with many actively growing defects is a physical distributed system, which transits from one physical state to another. This transition finally leads to failure of its components, each component being a defect. Such process can be described by a Markov process. The degradation of the PS (measured as monotonous deterioration of its failure pressure Pf (t)) is considered as a non-homogeneous pure death Markov process (NPDMP) of the continuous time and discrete states type. Failure pressure is calculated using one of the internationally recognized pipeline design codes: B13G, B31Gmod, DNV, Battelle and Shell-92. The NPDMP is described by a system of non-homogeneous differential equations, which allows calculating the probability of defects failure pressure being in each of its possible states. On the basis of these probabilities the gamma-percent residual life of defects is calculated. In other words, the moment of time tγ is calculated, which is a random variable, when the failure pressure of pipeline defect Pf (tγ) > Pop, with probability γ, where Pop is the operating pressure. The developed methodology was successfully applied to a real life case, which is presented and discussed.

Moment-Based Hybrid Polynomial Chaos Method for Interval and Random Uncertain Analysis of Periodical Composite Structural-Acoustic System with Multi-Scale Parameters

2020 ◽  
pp. 2050041
Author(s):  
Ning Chen ◽  
Jiaojiao Chen ◽  
Shengwen Yin
Keyword(s):  
Polynomial Chaos ◽  
Random Variable ◽  
Acoustic System ◽  
Passenger Compartment ◽  
Scale Parameters ◽  
Multi Scale ◽  
Arbitrary Polynomial ◽  
Random Moment ◽  
The Moment ◽  
Probability Density Function Pdf

An interval and random moment-based arbitrary polynomial chaos method (IRMAPCM) is proposed in this paper for the analysis of periodical composite structural-acoustic systems with multi-scale uncertain-but-bounded parameters. In IRMAPCM, the response of structural-acoustic system is approximated as moment-based arbitrary polynomial chaos (maPC) expansion. IRMAPCM can construct the polynomial basis according to the moment of the random variable without knowing the Probability Density Function (PDF), which can avoid the errors introduced by estimating the PDF. Numerical examples of a hexahedral box and an automobile passenger compartment are given to investigate the effectiveness of IRMAPCM for the prediction of the sound pressure response of structural-acoustic systems.

On the linear exponential family

1968 ◽  
Vol 64 (2) ◽  
pp. 481-483 ◽  
Author(s):  
J. K. Wani
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Scalar Product ◽  
Exponential Family ◽  
Characterization Theorem ◽  
Random Variable ◽  
Moment Generating Function ◽  
The Moment ◽  
Image Position

In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given bywhere a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

Moments for multidimensional Mandelbrot cascades

2016 ◽  
Vol 53 (1) ◽  
pp. 1-21
Author(s):  
Chunmao Huang
Keyword(s):  
Random Walk ◽  
Random Matrices ◽  
Random Variable ◽  
Tail Probability ◽  
Decay Rates ◽  
Branching Random Walk ◽  
Sufficient Condition ◽  
Random Vectors ◽  
Products Of Random Matrices ◽  
Sums Of Products

Abstract We consider the distributional equation Z =D ∑k=1NAkZ(k), where N is a random variable taking value in N0 = {0, 1, . . .}, A1, A2, . . . are p x p nonnegative random matrices, and Z, Z(1), Z(2), . . ., are independent and identically distributed random vectors in R+p with R+ = [0, ∞), which are independent of (N, A1, A2, . . .). Let {Yn} be the multidimensional Mandelbrot martingale defined as sums of products of random matrices indexed by nodes of a Galton–Watson tree plus an appropriate vector. Its limit Y is a solution of the equation above. For α > 1, we show a sufficient condition for E|Y|α ∈ (0, ∞). Then for a nondegenerate solution Z of the distributional equation above, we show the decay rates of Ee-t∙Z as |t| → ∞ and those of the tail probability P(y ∙ Z ≤ x) as x → 0 for given y = (y1, . . ., yp) ∈ R+p, and the existence of the harmonic moments of y ∙ Z. As an application, these results concerning the moments (of positive and negative orders) of Y are applied to a special multitype branching random walk. Moreover, for the case where all the vectors and matrices of the equation above are complex, a sufficient condition for the Lα convergence and the αth-moment of the Mandelbrot martingale {Yn} are also established.

On certain coloring parameters of Mycielski graphs of some graphs

2018 ◽  
Vol 10 (03) ◽  
pp. 1850030
Author(s):  
N. K. Sudev ◽  
K. P. Chithra ◽  
K. A. Germina ◽  
S. Satheesh ◽  
Johan Kok
Keyword(s):  
Graph Coloring ◽  
Random Variable ◽  
Proper Coloring ◽  
Discrete Random Variable ◽  
Variable Formula ◽  
Statistical Measures ◽  
Random Experiment ◽  
Mean And Variance

Coloring the vertices of a graph [Formula: see text] according to certain conditions can be considered as a random experiment and a discrete random variable [Formula: see text] can be defined as the number of vertices having a particular color in the proper coloring of [Formula: see text]. The concepts of mean and variance, two important statistical measures, have also been introduced to the theory of graph coloring and determined the values of these parameters for a number of standard graphs. In this paper, we discuss the coloring parameters of the Mycielskian of certain standard graphs.

Exact cash-account deflator for the G2++ model

2020 ◽  
Vol 07 (01) ◽  
pp. 2050009
Author(s):  
Francesco Strati ◽  
Luca G. Trussoni
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Insurance Industry ◽  
Random Variable ◽  
Time Span ◽  
Discrete Random Variable ◽  
Variable Formula ◽  
Monte Carlo Simulation Technique ◽  
Insurance Contracts ◽  
Embedded Options

In this paper, we shall propose a Monte Carlo simulation technique applied to a G2++ model: even when the number of simulated paths is small, our technique allows to find a precise simulated deflator. In particular, we shall study the transition law of the discrete random variable :[Formula: see text] in the time span [Formula: see text] conditional on the observation at time [Formula: see text], and we apply it in a recursive way to build the different paths of the simulation. We shall apply the proposed technique to the insurance industry, and in particular to the issue of pricing insurance contracts with embedded options and guarantees.

THE ASYMPTOTIC BEHAVIOR OF SEMI-INVARIANTS FOR LINEAR STOCHASTIC SYSTEMS

2002 ◽  
Vol 02 (02) ◽  
pp. 281-294
Author(s):  
G. N. MILSTEIN
Keyword(s):  
Asymptotic Behavior ◽  
Lyapunov Exponent ◽  
Linear System ◽  
Characteristic Function ◽  
Stochastic Systems ◽  
Random Variable ◽  
Characteristic Functions ◽  
Linear Stochastic Systems ◽  
Moment Lyapunov Exponent ◽  
The Moment

The asymptotic behavior of semi-invariants of the random variable ln |X(t,x)|, where X(t,x) is a solution of a linear system of stochastic differential equations, is connected with the moment Lyapunov exponent g(p). Namely, it is obtained that the nth semi-invariant is asymptotically proportional to the time t with the coefficient of proportionality g(n)(0). The proof is based on the concept of analytic characteristic functions. It is also shown that the asymptotic behavior of the analytic characteristic function of ln |X(t,x)| in a neighborhood of the origin of the complex plane is controlled by the extension g(iz) of g(p).

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