Complex Random Variables Research Articles

Almost sure behavior of the critical points of random polynomials

AbstractLet be a sequence of independent and identically distributed complex random variables with common distribution and let be the associated random polynomial in . Kabluchko established the conjecture stated by Pemantle and Rivin that the empirical measure associated with the critical points of converges weakly in probability to the base measure . In this note, we establish that the convergence, in fact, holds in the almost sure sense. Our result positively answers a question raised by Kabluchko and formalized as a conjecture in the recent paper (Michelen and Vu [arXiv:2212.11867]).

Oscillating neural circuits: Phase, amplitude, and the complex normal distribution.

Multiple oscillating time series are typically analyzed in the frequency domain, where coherence is usually said to represent the magnitude of the correlation between two signals at a particular frequency. The correlation being referenced is complex-valued and is similar to the real-valued Pearson correlation in some ways but not others. We discuss the dependence among oscillating series in the context of the multivariate complex normal distribution, which plays a role for vectors of complex random variables analogous to the usual multivariate normal distribution for vectors of real-valued random variables. We emphasize special cases that are valuable for the neural data we are interested in and provide new variations on existing results. We then introduce a complex latent variable model for narrowly band-pass-filtered signals at some frequency, and show that the resulting maximum likelihood estimate produces a latent coherence that is equivalent to the magnitude of the complex canonical correlation at the given frequency. We also derive an equivalence between partial coherence and the magnitude of complex partial correlation, at a given frequency. Our theoretical framework leads to interpretable results for an interesting multivariate dataset from the Allen Institute for Brain Science.

Knowledge Graph Embedding by Normalizing Flows

A key to knowledge graph embedding (KGE) is to choose a proper representation space, e.g., point-wise Euclidean space and complex vector space. In this paper, we propose a unified perspective of embedding and introduce uncertainty into KGE from the view of group theory. Our model can incorporate existing models (i.e., generality), ensure the computation is tractable (i.e., efficiency) and enjoy the expressive power of complex random variables (i.e., expressiveness). The core idea is that we embed entities/relations as elements of a symmetric group, i.e., permutations of a set. Permutations of different sets can reflect different properties of embedding. And the group operation of symmetric groups is easy to compute. In specific, we show that the embedding of many existing models, point vectors, can be seen as elements of a symmetric group. To reflect uncertainty, we first embed entities/relations as permutations of a set of random variables. A permutation can transform a simple random variable into a complex random variable for greater expressiveness, called a normalizing flow. We then define scoring functions by measuring the similarity of two normalizing flows, namely NFE. We construct several instantiating models and prove that they are able to learn logical rules. Experimental results demonstrate the effectiveness of introducing uncertainty and our model. The code is available at https://github.com/changyi7231/NFE.

Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices

We establish a quantitative version of the Tracy–Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix X∗X converge to its Tracy–Widom limit at a rate nearly N−1/3, where X is an M×N random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to infinity at a constant rate. This result improves the previous estimate N−2/9 obtained by Wang (2019). Our proof relies on a Green function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.

Aspects of the Kahane–Salem–Zygmund inequalities in Banach spaces

The main aim of this work is to discuss several different approaches to the celebrated Kahane–Salem–Zygmund inequalities. In particular, we prove estimates for exponential Orlicz norms of averages sup _{1le jle N}Big |sum _{i=1}^K a_i(j) gamma _iBig |,, where (a_i(j)) in ell _infty ^N, , 1 le i le K and the (gamma _i) form a sequence of real or complex subgaussian random variables. Lifting these inequalities to finite dimensional Banach spaces, we get some new Kahane–Salem–Zygmund type inequalities—in particular, for spaces of subgaussian random polynomials and multilinear forms on finite dimensional Banach spaces, and also for subgaussian random Dirichlet polynomials.

The bootstrap for testing the equality of two multivariate time series with an application to financial markets

The problem of testing the equality of the generating processes of two multivariate time series is addressed in this work. To this aim, we construct four tests based on a distance measure between stochastic processes. The metric is defined in terms of the quantile cross-spectral densities of both processes. A proper estimate of this dissimilarity is the cornerstone of the proposed tests. The first test employs the asymptotic distribution of the estimate, which we derive from some standard results on complex random variables and which is useful in its own right. The bad behaviour of this test when compared with alternative ones is shown. The three remaining techniques are based on the bootstrap. Specifically, a particular bootstrap method for spectral densities and extensions of the moving blocks bootstrap and the stationary bootstrap are used for their construction. The approaches are assessed in a broad range of scenarios under the null and the alternative hypothesis. The results from the analyses show that the procedure based on the stationary bootstrap exhibits the best overall performance in terms of both size and power. The proposed techniques are used to answer the question about whether or not the dotcom bubble crash of 2000s permanently impacted the global market behavior.

Generalized Maximum Complex Correntropy Augmented Adaptive IIR Filtering.

Augmented IIR filter adaptive algorithms have been considered in many studies, which are suitable for proper and improper complex-valued signals. However, lots of augmented IIR filter adaptive algorithms are developed under the mean square error (MSE) criterion. It is an ideal optimality criterion under Gaussian noises but fails to model the behavior of non-Gaussian noise found in practice. Complex correntropy has shown robustness under non-Gaussian noises in the design of adaptive filters as a similarity measure for the complex random variables. In this paper, we propose a new augmented IIR filter adaptive algorithm based on the generalized maximum complex correntropy criterion (GMCCC-AIIR), which employs the complex generalized Gaussian density function as the kernel function. Stability analysis provides the bound of learning rate. Simulation results verify its superiority.

Probabilistic solution of a homogeneous linear second-order differential equation with randomized complex coefficients

In this paper, an exact expression for the first probability density function of the solution stochastic process to a randomized homogeneous linear second-order complex differential equation is determined. To complete the probabilistic analysis, the first probability density functions of the real and complex contributions of the solution stochastic process are also calculated. To compute the densities, the random variable transformation method is applied under general hypothesis, all coefficients and initial conditions are absolutely continuous complex random variables. The capability of the theoretical results established is demonstrated by several numerical examples. Finally, we show the applicability of the method in engineering, by analysing the solution of a randomized simple harmonic oscillator, defined on the complex domain, and comparing our results with those obtained by using Monte Carlo simulations.

On impropriety for a large-sized discrete fourier transform of a real-valued stationary process

It is known that a finite-sized discrete Fourier transform (DFT) of a real-valued colored stationary random process results in improperly distributed complex random variables in the frequency domain. It is also well-known that a large-sized DFT of a colored random process approximates the behavior of the Karhunen–Loève (KL) transform, in terms of achieving approximate decorrelation between the frequency domain terms. We show that this decorrelating behavior also manifests itself in terms of the approximate complex circular (proper) nature of the majority of the transform domain terms, except near the band edges, where the nature of the complex distribution may be non-circular. We derive an analytical result relating the DFT size with the degree of impropriety, and also demonstrate the same via simulations. The result is important in the context of wideband xDSL (Digital Subscriber Line) discrete multi-tone (DMT) receivers affected by extraneous spectrally colored noise sources, besides being of general interest.

Some applications of the Menshov–Rademacher theorem

Given a sequence (X n ) of real or complex random variables and a sequence of numbers (a n ), an interesting problem is to determine the conditions under which the series ∑ n=1 ∞ a n X n is almost surely convergent. This paper extends the classical Menshov–Rademacher theorem on the convergence of orthogonal series to general series of dependent random variables and derives interesting sufficient conditions for the almost everywhere convergence of trigonometric series with respect to singular measures whose Fourier transform decays to 0 at infinity with positive rate.

A necessary and sufficient condition for convergence of the zeros of random polynomials

Consider random polynomials of the form Gn=∑i=0nξipi, where the ξi are i.i.d. non-degenerate complex random variables, and {pi} is a sequence of orthonormal polynomials with respect to a regular measure τ supported on a compact set K. We show that the zero measure of Gn converges weakly almost surely to the equilibrium measure of K if and only if Elog⁡(1+|ξ0|)<∞. This generalizes the corresponding result of Ibragimov and Zaporozhets in the case when pi(z)=zi. We also show that the zero measure of Gn converges weakly in probability to the equilibrium measure of K if and only if P(|ξ0|>en)=o(n−1).Our proofs rely on results from small ball probability and exploit the structure of general orthogonal polynomials. Our methods also work for sequences of asymptotically minimal polynomials in Lp(τ), where p∈(0,∞]. In particular, sequences of Lp-minimal polynomials and (normalized) Faber and Fekete polynomials fall into this class.

A note on the universality of ESDs of inhomogeneous random matrices

In this short note, we extend the celebrated results of Tao and Vu, and Krishnapur on the universality of empirical spectral distributions to a wide class of inhomogeneous complex random matrices, by showing that a technical and hard-to-verify Fourier domination assumption may be replaced simply by a natural uniform anti-concentration assumption. Along the way, we show that inhomogeneous complex random matrices, whose expected squared Hilbert-Schmidt norm is quadratic in the dimension, and whose entries (after symmetrization) are uniformly anti-concentrated at $0$ and infinity, typically have smallest singular value $\Omega(n^{-1/2})$. The rate $n^{-1/2}$ is sharp, and closes a gap in the literature. Our proofs closely follow recent works of Livshyts, and Livshyts, Tikhomirov, and Vershynin on inhomogeneous real random matrices. The new ingredient is a couple of anti-concentration inequalities for sums of independent, but not necessarily identically distributed, complex random variables, which may also be useful in other contexts.

First-order linear differential equations whose data are complex random variables: Probabilistic solution and stability analysis via densities

&lt;abstract&gt;&lt;p&gt;Random initial value problems to non-homogeneous first-order linear differential equations with complex coefficients are probabilistically solved by computing the first probability density of the solution. For the sake of generality, coefficients and initial condition are assumed to be absolutely continuous complex random variables with an arbitrary joint probability density function. The probability of stability, as well as the density of the equilibrium point, are explicitly determined. The Random Variable Transformation technique is extensively utilized to conduct the overall analysis. Several examples are included to illustrate all the theoretical findings.&lt;/p&gt;&lt;/abstract&gt;

On the Skitovich–Darmois theorem for complex and quaternion random variables

Abstract We prove the following theorem. Let α = a + i ⁢ b {\alpha=a+ib} be a nonzero complex number. Then the following statements hold: (i) Let either b ≠ 0 {b\neq 0} or b = 0 {b=0} and a &gt; 0 {a&gt;0} . Let ξ 1 {\xi_{1}} and ξ 2 {\xi_{2}} be independent complex random variables. Assume that the linear forms L 1 = ξ 1 + ξ 2 {L_{1}=\xi_{1}+\xi_{2}} and L 2 = ξ 1 + α ⁢ ξ 2 {L_{2}=\xi_{1}+\alpha\xi_{2}} are independent. Then ξ j {\xi_{j}} are degenerate random variables. (ii) Let b = 0 {b=0} and a &lt; 0 {a&lt;0} . Then there exist complex Gaussian random variables in the wide sense ξ 1 {\xi_{1}} and ξ 2 {\xi_{2}} such that they are not complex Gaussian random variables in the narrow sense, whereas the linear forms L 1 = ξ 1 + ξ 2 {L_{1}=\xi_{1}+\xi_{2}} and L 2 = ξ 1 + α ⁢ ξ 2 {L_{2}=\xi_{1}+\alpha\xi_{2}} are independent.

Zeros of Gaussian power series with dependent random variables

Peres and Virág in 2005 studied the zeros of the random power series ∑ n = 1 ∞ a n z n − 1 in the unit disc whose coefficients ( a n ) are independent and identically distributed symmetrical Gaussian complex random variables. They discovered that its zeros constitute a determinant point process and derived many other interesting related properties. In this paper, we study the general Gaussian power series where the independence requirement is relaxed by taking f ( z ) = ∑ n = 1 ∞ Δ n z n , where ( Δ n ) is the classical fractional Gaussian noise of index 0 ≤ H < 1 . The intensity of the point process of the zeros of f is used to derive important properties. As for the case of independent variables (corresponding to H = 1 / 2 ), the zeros cluster around the unit circle, but in any domain inside the open unit disc, the dependent case (that is, H ≠ 1 / 2 ) yields fewer zeros than the classical case of independent random variables. This implies that the clustering around the unit circle is faster than in the independent case. Moreover, in contrast to independent random variables, the zeros of f are generally asymmetrically distributed in the unit circle. However, asymptotically, the two point processes are equivalent except near the sole point z = 1 . An interesting open problem is whether this point process is also a determinantal process.

On The Quotient of a Centralized and a Non-centralized Complex Gaussian Random Variable.

A detailed investigation of the quotient of two independent complex random variables is presented. The numerator has a zero mean, and the denominator has a non-zero mean. A normalization step is taken prior to the theoretical developments in order to simplify the formulation. Next, an indirect approach is taken to derive the statistics of the modulus and phase angle of the quotient. That in turn enables a straightforward extension of the statistical results to real and imaginary parts. After the normalization procedure, the probability density function of the quotient is found as a function of only the mean of the random variable that corresponds to the denominator term. Asymptotic analysis shows that the quotient closely resembles a normally-distributed complex random variable as the mean becomes large. In addition, the first and second moments, as well as the approximate of the second moment of the clipped random variable, are derived, which are closely related to practical applications in complex-signal processing such as microwave metrology of scattering-parameters. Tolerance intervals associated with the ratio of complex random variables are presented.

Оцiнки розподiлу напiвнорм Гельдера вiд дiйсних стацiонарних гауссових процесiв зi стiйкою кореляцiйною функцiєю

Complex random variables and processes with a vanishing pseudo-correlation are called proper. There is a class of stationary proper complex random processes that have a stable correlation function. In the present article we consider real stationary Gaussian processes with a stable correlation function. It is shown that the trajectories of stationary Gaussian proper complex random processes with zero mean belong to the Orlich space generated by the function $U(x) = e^{x^2/2}-1$. Estimates are obtained for the distribution of semi-norms of sample functions of Gaussian proper complex random processes with a stable correlation function, defined on the compact $\mathbb{T} = [0,T]$, in Hölder spaces.

Eigenvalues of large chiral non-Hermitian random matrices

We study a non-Hermitian chiral random matrix of which the eigenvalues are complex random variables. The empirical distributions and the radius of the eigenvalues are investigated. The limit of the empirical distributions is a new probability distribution defined on the complex plane. The graphs of the density functions are plotted; the surfaces formed by the density functions are understood through their convexity and their Gaussian curvatures. The limit of the radius is a Gumbel distribution. The main observation is that the joint density function of the eigenvalues of the chiral ensemble, after a transformation, becomes a rotation-invariant determinantal point process on the complex plane. Then, the eigenvalues are studied by the tools developed by Jiang and Qi [J. Theor. Probab. 30, 326 (2017); 32, 353 (2019)]. Most efforts are devoted to deriving the central limit theorems for distributions defined by the Bessel functions via the method of steepest descent and the estimates of the zero of a non-trivial equation as the saddle point.

A Journey From Improper Gaussian Signaling to Asymmetric Signaling

The deviation of continuous and discrete complex random variables from the traditional proper and symmetric assumption to a generalized improper and asymmetric characterization (accounting correlation between a random entity and its complex conjugate), respectively, introduces new design freedom and various potential merits. As such, the theory of impropriety has vast applications in medicine, geology, acoustics, optics, image and pattern recognition, computer vision, and other numerous research fields with our main focus on the communication systems. The journey begins from the design of improper Gaussian signaling in the interference-limited communications and leads to a more elaborate and practically feasible asymmetric discrete modulation design. Such asymmetric shaping bridges the gap between theoretically and practically achievable limits with sophisticated transceiver and detection schemes in both coded/uncoded wireless/optical communication systems. Interestingly, introducing asymmetry and adjusting the transmission parameters according to some design criterion render optimal performance without affecting the bandwidth or power requirements of the systems. This dual-flavored article initially presents the tutorial base content covering the interplay of reality/complexity, propriety/impropriety and circularity/noncircularity and then surveys majority of the contributions in this enormous journey.

A zero-cumulant random variable and its applications

High Order Statistics (HOS) are widely used in many algorithms ranging from blind identification to signal separation. A well-known identifiability result is that at most one Gaussian source should be present in static mixtures [J. Eriksson, V. Koivunen, Identifiability, separability and uniqueness of linear ICA models, IEEE Signal Process. Lett. (2004) 601-604]. The reason for this is that these algorithms utilize cumulants of order higher than two, and that they are all null for circular Gaussian random variables [M. Kendall, A. Stuart, The Advanced Theory of Statistics, Distribution Theory, vol. 1, C. Griffin, 1977]. Simple examples of non-Gaussian complex random variables having zero cumulants of order three to seven are given, which can be encountered in the real world.