Complex-valued Random Variables Research Articles

Arbitrary Random Variables and Wiman’s Inequality

We study the class of random entire functions given by power series, in which the coefficients are formed as the product of an arbitrary sequence of complex numbers and two sequences of random variables. One of them is the Rademacher sequence, and the other is an arbitrary complex-valued sequence from the class of sequences of random variables, determined by a certain restriction on the growth of absolute moments of a fixed degree from the maximum of the module of each finite subset of random variables. In the paper we prove sharp Wiman–Valiron’s type inequality for such random entire functions, which for given p∈(0;1) holds with a probability p outside some set of finite logarithmic measure. We also considered another class of random entire functions given by power series with coefficients, which, as above, are pairwise products of the elements of an arbitrary sequence of complex numbers and a sequence of complex-valued random variables described above. In this case, similar new statements about not improvable inequalities are also obtained.

Random matrices associated to Young diagrams

We consider the singular values of certain Young diagram shaped random matrices. For block-shaped random matrices, the empirical distribution of the squares of the singular eigenvalues converges almost surely to a distribution whose moments are a generalization of the Catalan numbers. The limiting distribution is the density of a product of rescaled independent Beta random variables and its Stieltjes–Cauchy transform has a hypergeometric representation. In special cases we recover the Marchenko–Pastur and Dykema–Haagerup measures of square and triangular random matrices, respectively. We find a further factorization of the moments in terms of two complex-valued random variables that generalizes the factorization of the Marchenko–Pastur law as product of independent uniform and arcsine random variables.

ПРОСТРАНСТВА ДИФФЕРЕНЦИАЛЬНЫХ ФОРМ СО СТОХАСТИЧЕСКИМИ КОМПЛЕКСНОЗНАЧНЫМИ КОЭФФИЦИЕНТАМИ

This article investigates the construction of spaces of differential forms with coefficients which are stochastic complex-valued K-processes. A complete probability space and complex-valued random variables on measurable subsets of this space are considered, and continuous random complexvalued K- processes are also introduced. Next, we construct spaces of differential forms with coefficients in the form of such stochastic complex-valued K-processes.

Non-probabilistic uncertainty quantification for dynamic characterization functions using complex ratio interval arithmetic operation of multidimensional parallelepiped model

Uncertainty quantification for the experimental estimations of dynamic characterization functions, including frequency response functions (FRFs) and transmissibility functions (TFs), is of practical importance in improving the robustness of the real applications of these functions for system identification and structural health monitoring. Interval analysis is an appealing tool for dealing with the uncertainties of engineering problems in which only the bounds of uncertain parameters are available. FRFs and TFs are complex-valued random variables. However, due to the negligence of the dependencies of complex-valued variables, the existing complex ratio interval arithmetic operation can be overly conservative. In this study, the polar representation of complex ratio numbers was extended to complex ratio polar intervals and a multidimensional parallelepiped (MP) interval model was introduced to accommodate the dependence between the numerator and the denominator. Based on the explicit expressions of the MP model through a dependence matrix, two new global extrema searching schemes with and without the regularization of the uncertainty domain of the MP model were proposed in order to derive the explicit formulas of the upper and lower bounds of the magnitudes and phases of the FRFs and TFs. The new schemes were then applied to the uncertainty propagation for a numerically simulated beam and a bridge subjected to a single excitation. The results showed that the interval overestimation problem could be significantly alleviated by using the new complex-valued ratio interval arithmetic operation of the parallelepiped model.

The Variance of the Number of Zeros for Complex Random Polynomials Spanned by OPUC

Let $$\{\varphi _k\}_{k=0}^\infty $$ be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure $$ \mu $$. We study the variance of the number of zeros of random linear combinations of the form $$\begin{aligned} P_n(z)=\sum _{k=0}^{n}\eta _k\varphi _k(z), \end{aligned}$$where $$\{\eta _k\}_{k=0}^n $$ are complex-valued random variables. Under the assumption that the distribution for each $$\eta _k$$ satisfies certain uniform bounds for the fractional and logarithmic moments, for the cases when $$\{\varphi _k\}$$ are regular in the sense of Ullman–Stahl–Totik or are such that the measure of orthogonality $$\mu $$ satisfies $$d\mu (\theta )=w(\theta )d\theta $$ where $$w(\theta )=v(\theta )\prod _{j=1}^J|\theta - \theta _j|^{\alpha _j}$$, with $$v(\theta )\ge c>0$$, $$\theta ,\theta _j\in [0,2\pi )$$, and $$\alpha _j>0$$, we give a quantitative estimate on the variance of the number of zeros of $$P_n$$ in sectors that intersect the unit circle. When $$\{\varphi _k\}$$ are real-valued on the real-line from the Nevai class and $$\{\eta _k\}$$ are i.i.d. complex-valued standard Gaussian, we obtain a formula for the limiting value of variance of the number of zeros of $$P_n$$ in annuli that do not contain the unit circle.

On the local pairing behavior of critical points and roots of random polynomials

We study the pairing between zeros and critical points of the polynomial $p_{n}(z) = \prod _{j=1}^{n}(z-X_{j})$, whose roots $X_{1}, \ldots , X_{n}$ are complex-valued random variables. Under a regularity assumption, we show that if the roots are independent and identically distributed, the Wasserstein distance between the empirical distributions of roots and critical points of $p_{n}$ is on the order of $1/n$, up to logarithmic corrections. The proof relies on a careful construction of disjoint random Jordan curves in the complex plane, which allow us to naturally pair roots and nearby critical points. In addition, we establish asymptotic expansions to order $1/n^{2}$ for the locations of the nearest critical points to several fixed roots. This allows us to describe the joint limiting fluctuations of the critical points as $n$ tends to infinity, extending a recent result of Kabluchko and Seidel. Finally, we present a local law that describes the behavior of the critical points when the roots are neither independent nor identically distributed.

Entropy Bounds on Abelian Groups and the Ruzsa Divergence

Over the past few years, a family of interesting new inequalities for the entropies of sums and differences of random variables has been developed by Ruzsa, Tao and others, motivated by analogous results in additive combinatorics. The present work extends these earlier results to the case of random variables taking values in $\mathbb{R}^n$ or, more generally, in arbitrary locally compact and Polish abelian groups. We isolate and study a key quantity, the Ruzsa divergence between two probability distributions, and we show that its properties can be used to extend the earlier inequalities to the present general setting. The new results established include several variations on the theme that the entropies of the sum and the difference of two independent random variables severely constrain each other. Although the setting is quite general, the result are already of interest (and new) for random vectors in $\mathbb{R}^n$. In that special case, quantitative bounds are provided for the stability of the equality conditions in the entropy power inequality; a reverse entropy power inequality for log-concave random vectors is proved; an information-theoretic analog of the Rogers-Shephard inequality for convex bodies is established; and it is observed that some of these results lead to new inequalities for the determinants of positive-definite matrices. Moreover, by considering the multiplicative subgroups of the complex plane, one obtains new inequalities for the differential entropies of products and ratios of nonzero, complex-valued random variables.

Equidistribution, uniform distribution: a probabilist’s perspective

The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. A motivated uninitiated peer could encounter difficulties perusing the literature, due to various synonyms and polysemes used by different schools. One purpose of this note is to provide a short introduction for probabilists. We proceed by recalling a perspective originating in a work of the second author from 2002. Using it, various new examples of completely uniformly distributed $\mathsf{mod}~1$ sequences, in the “metric” (meaning almost sure stochastic) sense, can be easily exhibited. In particular, we point out natural generalizations of the original $p$-multiply equidistributed sequence $k^{p}\,t\ \mathsf{mod}~1$, $k\geq1$ (where $p\in\mathbb{N}$ and $t\in[0,1]$), due to Hermann Weyl in 1916. In passing, we also derive a Weyl-like criterion for weakly completely equidistributed (also known as WCUD) sequences, of substantial recent interest in MCMC simulations. The translation from number theory to probability language brings into focus a version of the strong law of large numbers for weakly correlated complex-valued random variables, the study of which was initiated by Weyl in the aforementioned manuscript, followed up by Davenport, Erdős and LeVeque in 1963, and greatly extended by Russell Lyons in 1988. In this context, an application to $\infty$-distributed Koksma’s numbers $t^{k}\ \mathsf{mod}~1$, $k\geq1$ (where $t\in[1,a]$ for some $a>1$), and an important generalization by Niederreiter and Tichy from 1985 are discussed. The paper contains negligible amount of new mathematics in the strict sense, but its perspective and open questions included in the end could be of considerable interest to probabilists and statisticians, as well as certain computer scientists and number theorists.

Complex obtuse random walks and their continuous-time limits

We study a particular class of complex-valued random variables and their associated random walks: the complex obtuse random variables. They are the generalization to the complex case of the real-valued obtuse random variables which were introduced in \cite{A-E} in order to understand the structure of normal martingales in $\RR^n$.The extension to the complex case is mainly motivated by considerations from Quantum Statistical Mechanics, in particular for the seek of a characterization of those quantum baths acting as classical noises. The extension of obtuse random variables to the complex case is far from obvious and hides very interesting algebraical structures. We show that complex obtuse random variables are characterized by a 3-tensor which admits certain symmetries which we show to be the exact 3-tensor analogue of the normal character for 2-tensors (i.e. matrices), that is, a necessary and sufficient condition for being diagonalizable in some orthonormal basis. We discuss the passage to the continuous-time limit for these random walks and show that they converge in distribution to normal martingales in $\CC^N$. We show that the 3-tensor associated to these normal martingales encodes their behavior, in particular the diagonalization directions of the 3-tensor indicate the directions of the space where the martingale behaves like a diffusion and those where it behaves like a Poisson process. We finally prove the convergence, in the continuous-time limit, of the corresponding multiplication operators on the canonical Fock space, with an explicit expression in terms of the associated 3-tensor again.

Random Polynomials and Pluripotential-Theoretic Extremal Functions

There is a natural pluripotential-theoretic extremal function V K,Q associated to a closed subset K of $\mathbb {C}^{m}$ and a real-valued, continuous function Q on K. We define random polynomials H n whose coefficients with respect to a related orthonormal basis are independent, identically distributed complex-valued random variables having a very general distribution (which includes both normalized complex and real Gaussian distributions) and we prove results on a.s. convergence of a sequence $\{\frac {1}{n}\log |H_{n}|\}$ pointwise and in $L^{1}_{loc}(\mathbb {C}^{m})$ to V K,Q . In addition we obtain results on a.s. convergence of a sequence of normalized zero currents $\{dd^{c}\left (\frac {1}{n}\log |H_{n}|\right )\}$ to d d c V K,Q as well as asymptotics of expectations of these currents. All these results extend to random polynomial mappings and to a more general setting of positive holomorphic line bundles over a compact Kähler manifold.

A bivariate CLT under rho-prime mixing

The main result is a bivariate central limit theorem for “rectangular” sums of dependent complex-valued random variables, indexed by $\mathbb{Z} ^{d}$, which are $\rho^{\prime}$-mixing. An interesting corollary concerning the limiting behavior of the moments of the sums is then proved.

Generalized generating function with tucker decomposition and alternating least squares for underdetermined blind identification

Generating function (GF) has been used in blind identification for real-valued signals. In this paper, the definition of GF is first generalized for complex-valued random variables in order to exploit the statistical information carried on complex signals in a more effective way. Then an algebraic structure is proposed to identify the mixing matrix from underdetermined mixtures using the generalized generating function (GGF). Two methods, namely GGF-ALS and GGF-TALS, are developed for this purpose. In the GGF-ALS method, the mixing matrix is estimated by the decomposition of the tensor constructed from the Hessian matrices of the GGF of the observations, using an alternating least squares (ALS) algorithm. The GGF-TALS method is an improved version of the GGF-ALS algorithm based on Tucker decomposition. More specifically, the original tensor, as formed in GGF-ALS, is first converted to a lower-rank core tensor using the Tucker decomposition, where the factors are obtained by the left singular-value decomposition of the original tensor’s mode-3 matrix. Then the mixing matrix is estimated by decomposing the core tensor with the ALS algorithm. Simulation results show that (a) the proposed GGF-ALS and GGF-TALS approaches have almost the same performance in terms of the relative errors, whereas the GGF-TALS has much lower computational complexity, and (b) the proposed GGF algorithms have superior performance to the latest GF-based baseline approaches.

On the asymptotic distribution of GLR for impropriety of complex signals

In this paper, the problem of testing impropriety (i.e., second-order noncircularity) of a sequence of complex-valued random variables (RVs) based on the generalized likelihood ratio test (GLRT) for Gaussian distributions is considered. Asymptotic (w.r.t. the data length) distributions of the GLR are given under the hypothesis that RVs are proper or improper, and under the true, not necessarily Gaussian distribution of the RVs. The considered RVs are independent but not necessarily identically distributed: assumption which has never been considered until now. This enables us to deal with the practical important situations of noncircular RVs disturbed by residual frequency offsets and additive circular noise. The receiver operating characteristic (ROC) of this test is derived as byproduct, an issue previously overlooked. Finally illustrative examples are presented in order to strengthen the obtained theoretical results.

A Universality Property of Gaussian Analytic Functions

We consider random analytic functions defined on the unit disk of the complex plane \(f(z) = \sum_{n=0}^{\infty} a_{n} X_{n} z^{n}\), where the Xn’s are i.i.d., complex-valued random variables with mean zero and unit variance. The coefficients an are chosen so that f(z) is defined on a domain of ℂ carrying a planar or hyperbolic geometry, and \(\mathbf{E}f(z)\overline{f(w)}\) is covariant with respect to the isometry group. The corresponding Gaussian analytic functions have been much studied, and their zero sets have been considered in detail in a monograph by Hough, Krishnapur, Peres, and Virag. We show that for non-Gaussian coefficients, the zero set converges in distribution to that of the Gaussian analytic functions as one transports isometrically to the boundary of the domain. The proof is elementary and general.

On Gaussian Product Modulus Distribution

In this letter, we derive a probability density function (PDF) for a modulus of product of two complex-valued Gaussian random variables. The PDF is expressed using Modified Bessel Functions, and the probability distribution is named Gaussian Product Modulus Distribution. Some examples of expectation calculation are provided.

Random Matrices: the Distribution of the Smallest Singular Values

Let ξ be a real-valued random variable of mean zero and variance 1. Let M n (ξ) denote the n × n random matrix whose entries are iid copies of ξ and σ n (M n (ξ)) denote the least singular value of M n (ξ). The quantity σ n (M n (ξ))2 is thus the least eigenvalue of the Wishart matrix $${M_nM_n^\ast}$$ . We show that (under a finite moment assumption) the probability distribution n σ n (M n (ξ))2 is universal in the sense that it does not depend on the distribution of ξ. In particular, it converges to the same limiting distribution as in the special case when ξ is real gaussian. (The limiting distribution was computed explicitly in this case by Edelman.) We also proved a similar result for complex-valued random variables of mean zero, with real and imaginary parts having variance 1/2 and covariance zero. Similar results are also obtained for the joint distribution of the bottom k singular values of M n (ξ) for any fixed k (or even for k growing as a small power of n) and for rectangular matrices. Our approach is motivated by the general idea of “property testing” from combinatorics and theoretical computer science. This seems to be a new approach in the study of spectra of random matrices and combines tools from various areas of mathematics.

Dicussion on the Meeting on ‘Statistical Approaches to Inverse Problems’

Johnstone, Kerkyacharian, Picard and Raimondo Johnstone, Kerkyacharian, Picard and Raimondo are interested in the inverse problem of estimating f where f has been convolved with g and then contaminated with white noise. This popular problem has been tackled by a wide variety of procedures and wavelet methods have recently generated considerable interest. Donoho’s (1995) seminal wavelet–vaguelette paper introduced the notion that wavelets would be a good choice for the representation of f since real life objects, such as images, are more likely to be efficiently represented using wavelets when compared with, for example, Fourier representations. Johnstone and his colleagues have moved the field on significantly. In particular, their procedure is more direct than wavelet–vaguelette or Abramovich and Silverman’s (1998) vaguelette–wavelet method; it can handle boxcar blur theoretically and practically, they have rates of convergence forp =2 (p defines the type of loss) and the paper innovates through use of the new maxiset approach. For me, the most appealing of these innovations is that of enabling the treatment of boxcar blur which is one of the most common types of inverse problem. However, is it really, really, the case that for rational a nothing can be done? Formula (4) compels us to say no, nothing can, but naively it still feels wrong. Formula (19) is the popular ‘signal-plus-noise’ model but here it is a little different from what normally appears in the literature because the quantities are complex-valued random variables. More specifically, the zl are zero-mean Gaussian variables which are complex valued and satisfy E.zlzk/= δlk. One question is why threshold the βk and not the yl directly? The covariance of the βk is given by cov.βk, βl/=n ∑ m ΨkmΨ l m

Random variables determined by the distributions of their digits in a numeration system with complex base

We study the distributions of complex-valued random variables determined by the distributions of their digits in a numeration system with complex base. We establish sufficient conditions for the singularity of such random variables, in particular, in the cases where their spectrum has Lebesgue measure zero (C-type singular distribution) or is a rectangle (S-type singular distribution).

One class of singular complex-valued random variables of the Jessen-Wintner type

We study the structure of the distribution of a complex-valued random variable ξ = Σa k ξ k , where ξ k are independent complex-valued random variables with discrete distribution and a k are terms of an absolutely convergent series. We establish a criterion of discreteness and sufficient conditions for singularity of the distribution of ξ and investigate the fractal properties of the spectrum.

C75. Path analysis for complex-valued random variables with an application to spectral analysis of time series

C75. Path analysis for complex-valued random variables with an application to spectral analysis of time series