Independent Gaussian Random Variables Research Articles

Multiplier subsample bootstrap for statistics of time series

Block-based bootstrap, block-based subsampling and multiplier bootstrap are three common nonparametric tools for statistical inference under dependent observations. Combining the ideas of those three, a novel resampling approach, the multiplier subsample bootstrap (MSB), is proposed. Instead of generating a resample from the observations, the MSB imitates the statistic by weighting the block-based subsample statistics with independent standard Gaussian random variables. Given the asymptotic normality of the statistic, the bootstrap validity is established under some mild moment conditions. Involving the idea of MSB, the other resampling approach, the hybrid multiplier subsampling periodogram bootstrap (HMP), is developed for mimicking frequency-domain spectral mean statistics in the paper. A simulation study demonstrates that both the MSB and HMP achieve good performance.

RIN Reduction in Gain-Switched InAs-InP(113)B Quantum Dot Laser Based on Multi-Population Rate Equations

For the first time, relative intensity noise (RIN) of an InAs-InP(113)B quantum dot (Q-Dot) laser is examined theoretically under the gain-switching condition with the application of a Gaussian pulse beam (GPB) to excited state (Exs) of the laser. The multi-population rate equations considering nonlinear gain are solved by the Runge–Kutta method. Noises are added to rate equations as Langevin noise sources and a different method is applied here for the addition of these noise sources to equations. In this method, new rate equations are defined to eliminate the cross-correlations between the noise sources to make them independent and simulate with independent white Gaussian random variables. Obtained results showed that RIN decreases with the increasing ac peak current and the increasing inhomogeneous broadening. It was also observed that further RIN reduction of about 30 dB/Hz and gain-switched short pulses with a high power due to Exs emission being obtained with the application of GPB to Exs of the laser. Moreover, it was demonstrated that while carrier noise generated from Exs and ground state (Grs) affect the RIN spectrum, there is no effect of wetting layer (Wly) carrier noise.

Entire Gaussian Functions: Probability of Zeros Absence

In this paper, we consider a random entire function of the form f(z,ω)=∑n=0+∞εn(ω1)×ξn(ω2)fnzn, where (εn) is a sequence of independent Steinhaus random variables, (ξn) is the a sequence of independent standard complex Gaussian random variables, and a sequence of numbers fn∈C is such that lim¯n→+∞|fn|n=0 and #{n:fn≠0}=+∞. We investigate asymptotic estimates of the probability P0(r)=P{ω:f(z,ω) has no zeros inside rD} as r→+∞ outside of some set E of finite logarithmic measure, i.e., ∫E∩[1,+∞)dlnr<+∞. The obtained asymptotic estimates for the probability of the absence of zeros for entire Gaussian functions are in a certain sense the best possible result. Furthermore, we give an answer to an open question of A. Nishry for such random functions.

An asymptotic approach to proving sufficiency of Stein characterisations

In extending Stein's method to new target distributions, the first step is to find a Stein operator that suitably characterises the target distribution. In this paper, we introduce a widely applicable technique for proving sufficiency of these Stein characterisations, which can be applied when the Stein operators are linear differential operators with polynomial coefficients. The approach involves performing an asymptotic analysis to prove that only one characteristic function satisfies a certain differential equation associated to the Stein characterisation. We use this approach to prove that all Stein operators with linear coefficients characterise their target distribution, and verify on a case-by-case basis that all polynomial Stein operators in the literature with coefficients of degree at most two are characterising. For $X$ denoting a standard Gaussian random variable and $H_p$ the $p$-th Hermite polynomial, we also prove, amongst other examples, that the Stein operators for $H_p(X)$, $p=3,4,\ldots,8$, with coefficients of minimal possible degree characterise their target distribution, and that the Stein operators for the products of $p=3,4,\ldots,8$ independent standard Gaussian random variables are characterising (in both settings the Stein operators for the cases $p=1,2$ are already known to be characterising). We leverage our Stein characterisations of $H_3(X)$ and $H_4(X)$ to derive characterisations of these target distributions in terms of iterated Gamma operators from Malliavin calculus, that are natural in the context of the Malliavin-Stein method.

Probability Distributions for Track Fitting and the Advanced Lucky Model

Probability distributions for the center of gravity are fundamental tools for track fitting. The center of gravity is a widespread algorithm for position reconstruction in tracker detectors for particle physics. Its standard use is always accompanied by an easy guess (Gaussian) for the probability distribution of the positioning errors. This incorrect assumption degrades the results of the fit. The explicit error forms evident Cauchy–Agnesi tails that render the use of variance minimizations problematic. Therefore, it is important to report probability distributions for some combinations of random variables essential for track fitting: x=ξ/(ξ+μ), y=(ξ−μ)/[2(ξ+μ)], w=ξ/μ, x=θ(x3−x1)(−x3)/(x3+x2)+θ(x1−x3)x1/(x1+x2) and x=(x1−x3)/(x1+x2+x3). The first two are partial forms of the two strip center of gravity. The fourth is the complete two strip center of gravity, and the fifth is a partial form of the three strip center of gravity. For the complexity of the forth equation, only approximate expressions of the probability are allowed. Analytical expressions are calculated assuming ξ, μ, x1, x2 and x3 independent Gaussian random variables. The analytical form of the probability for the two strip center of gravity allows one to construct an approximate proof for the lucky model of our previous paper. This proof also suggests how to complete the lucky model by its absence of a scaling constant, relevant to combine different detector types. This advanced lucky model (the super-lucky model) can be directly used in trackers composed of non-identical detectors. The construction of the super-lucky model is very simple. Simulations with this upgraded tool also show resolution improvements for a combination of two types of very different detectors, near to the resolutions of the schematic model.

Delocalization and Quantum Diffusion of Random Band Matrices in High Dimensions II: T-expansion

We consider Green's functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(\mathbb Z/L\mathbb Z)^d$. The entries $h_{xy}=\bar h_{yx}$ of $H$ are independent centered complex Gaussian random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. The variances satisfy a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. For any $n\in \mathbb N$, we construct an expansion of the $T$-variable, $T_{xy}=|m|^2 \sum_{\alpha}s_{x\alpha}|G_{\alpha y}|^2$, with an error $O(W^{-nd/2})$, and use it to prove a local law on the Green's function. This $T$-expansion was the main tool to prove the delocalization and quantum diffusion of random band matrices for dimensions $d\ge 8$ in part I of this series.

Two problems on random analytic functions in Fock spaces

AbstractLet $f(z)=\sum _{n=0}^\infty a_n z^n$ be an entire function on the complex plane, and let ${\mathcal R} f(z) = \sum _{n=0}^\infty a_n X_n z^n$ be its randomization induced by a standard sequence $(X_n)_n$ of independent Bernoulli, Steinhaus, or Gaussian random variables. In this paper, we characterize those functions $f(z)$ such that ${\mathcal R} f(z)$ is almost surely in the Fock space ${\mathcal F}_{\alpha }^p$ for any $p, \alpha \in (0,\infty )$ . Then such a characterization, together with embedding theorems which are of independent interests, is used to obtain a Littlewood-type theorem, also known as regularity improvement under randomization within the scale of Fock spaces. Other results obtained in this paper include: (a) a characterization of random analytic functions in the mixed-norm space ${\mathcal F}(\infty , q, \alpha )$ , an endpoint version of Fock spaces, via entropy integrals; (b) a complete description of random lacunary elements in Fock spaces; and (c) a complete description of random multipliers between different Fock spaces.

An efficient and spectral accurate numerical method for computing SDE driven by multivariate Gaussian variables

There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

Optimization of the mean-square approximation procedures for iterated Stratonovich stochastic integrals of multiplicities 1 to 3 with respect to components of the multi-dimensional Wiener process based on Multiple Fourier–Legendre series

The article is devoted to approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 3 by the method of multiple Fourier–Legendre series. The mentioned stochastic integrals are part of strong numerical methods with convergence order 1.5 for Ito stochastic differential equations with multidimensional noncommutative noise. These numerical methods are based on the so-called Taylor–Ito and Taylor–Stratonovich expansions. We calculate the exact lengths of sequences of independent standard Gaussian random variables required for the mean-square approximation of iterated Stratonovich stochastic integrals. Thus, the computational cost for the implementation of numerical methods can be significantly reduced.

Thin-shell theory for rotationally invariant random simplices

For fixed functions G,H:[0,∞)→[0,∞), consider the rotationally invariant probability density on Rn of the form μn(ds)=1 ZnG(‖s‖2)e−nH(‖s‖2)ds. We show that when n is large, the Euclidean norm ‖Yn‖2 of a random vector Yn distributed according to μn satisfies a thin-shell property, in that its distribution is highly likely to concentrate around a value s0 minimizing a certain variational problem. Moreover, we show that the fluctuations of this modulus away from s0 have the order 1∕n and are approximately Gaussian when n is large. We apply these observations to rotationally invariant random simplices: the simplex whose vertices consist of the origin as well as independent random vectors Y1n,…,Ypn distributed according to μn, ultimately showing that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior. Our class of measures includes the Gaussian distribution, the beta distribution and the beta prime distribution on Rn, provided a generalizing and unifying setting for the objects considered in Grote-Kabluchko-Thäle [Limit theorems for random simplices in high dimensions, ALEA 16, 141–177 (2019)]. Finally, the volumes of random simplices may be related to the determinants of random matrices, and we use our methods with this correspondence to show that if An is an n×n random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants c0,c1∈(0,∞) and an absolute constant C∈(0,∞) such that sups∈RPlogdet(An)−log(n−1)!−c0 1 2logn+c1<s−∫−∞se−u2∕2du 2π<C log3∕2n, sharpening the 1∕log1∕3+o(1)n bound in Nguyen and Vu [Random matrices: Law of the determinant, Ann. Probab. 42(1) (2014), 146–167].

Optimization of the mean-square approximation procedures for iterated Ito stochastic integrals based on multiple Fourier-Legendre series

The article is devoted to optimization of the mean-square approximation procedures for iterated Ito stochastic integrals of multiplicities 1 to 4 based on multiple Fourier-Legendre series. The mentioned stochastic integrals are part of strong numerical methods with convergence orders 1.0, 1.5, and 2.0 for Ito stochastic differential equations with multidimensional non-commutative noise. We show that the lengths of sequences of independent standard Gaussian random variables required for the mean-square approximation of iterated Ito stochastic integrals can be significantly reduced without the loss of the mean-square accuracy of approximation for these stochastic integrals.

On the Number of Colored Balls from a Fixed Set in a Multi-Color Urn Scheme without Return

We consider random vectors that consist of numbers of colored balls belonging to a fixed value set in a multi-color urn scheme without return. We proved, that under certain conditions, random vectors, consisting of centered and normed elements of these vectors, converge in distribution to a random vector made up of independent Gaussian random variables with means 0 and variances of 1. We also obtained limit theorems for the functions of these random vectors. Applications of these theorems to estimate probabilities of type I errors of the $$\chi^{2}$$ -test and to estimate probabilities of type I errors and type II errors of some statistical tests are given.

The Two-Layer Hierarchical Distribution Model of Zeros of Riemann’s Zeta Function along the Critical Line

This article numerically analyzes the distribution of the zeros of Riemann’s zeta function along the critical line (CL). The zeros are distributed according to a hierarchical two-layered model, one deterministic, the other stochastic. Following a complex plane anamorphosis involving the Lambert function, the distribution of zeros along the transformed CL follows the realization of a stochastic process of regularly spaced independent Gaussian random variables, each linked to a zero. The value of the standard deviation allows the possible overlapping of adjacent realizations of the random variables, over a narrow confidence interval. The hierarchical model splits the ζ function into sequential equivalence classes, with the range of probability densities of realizations coinciding with the spectrum of behavioral styles of the classes. The model aims to express, on the CL, the coordinates of the alternating cancellations of the real and imaginary parts of the ζ function, to dissect the formula for the number of zeros below a threshold, to estimate the statistical laws of two consecutive zeros, of function maxima and moments. This also helps explain the absence of multiple roots.

A semicircle law and decorrelation phenomena for iterated Kolmogorov loops

We consider a standard one-dimensional Brownian motion on the time interval $[0,1]$ conditioned to have vanishing iterated time integrals up to order $N$. We show that the resulting processes can be expressed explicitly in terms of shifted Legendre polynomials and the original Brownian motion, and we use these representations to prove that the processes converge weakly as $N\to\infty$ to the zero process. This gives rise to a polynomial decomposition for Brownian motion. We further study the fluctuation processes obtained through scaling by $\sqrt{N}$ and show that they converge in finite dimensional distributions as $N\to\infty$ to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. The fluctuation result is a consequence of a limit theorem for Legendre polynomials which quantifies their completeness and orthogonality property. In the proof of the latter, we encounter a Catalan triangle.

Stochastic Modeling and Reliability Analysis of Wing Flutter

In this work, a physics-based first-order reliability method (FORM) algorithm is proposed for the flutter reliability analysis of an aircraft wing in the frequency domain. The limit state function, which is an implicit function of random variables, is defined in terms of the damping ratio of the aeroelastic system in a conditional sense on flow velocity. Two aeroelastic cases, namely, an airfoil section model and a cantilever wing model, are considered for carrying out the studies. These aeroelastic models have well separated mean bending and mean torsional modal frequencies. The geometric, structural, and aerodynamic parameters of airfoil and wing systems are modeled as independent Gaussian random variables. The effects of these on the statistics of frequency and damping ratio, and the cumulative distribution functions (CDFs) of flutter velocity are studied. In the case of the wing, the effects of modeling stiffness parameters as Gaussian random fields on the CDFs of flutter velocity are also studied. Here, spectral stochastic finite element method (SFEM) based on Karhunen–Loeve (K–L) expansion is used to discretize the random fields into random variables. From the study of an airfoil system, it is observed that parameters like torsional stiffness, elastic axis location, free stream density, and mass moment of inertia are more sensitive as compared with other parameters. However, in the case of the wing parameters such as torsional stiffness, free stream density, mass moment of inertia, and mass are observed to be more sensitive. The CDFs of flutter velocity obtained using the proposed algorithm are compared with Monte Carlo simulations (MCS) and found to be accurate. A comparative study of aeroelastic reliability for the wing is also carried out by treating stiffness parameters as random variables and random fields. It is observed that the CDFs of flutter velocity in the tail region are conservative when stiffness parameters are treated as random variables.

The real tau‐conjecture is true on average

Koiran's real τ‐conjecture claims that the number of real zeros of a structured polynomial given as a sum of m products of k real sparse polynomials, each with at most t monomials, is bounded by a polynomial in mkt. This conjecture has a major consequence in complexity theory since it would lead to superpolynomial lower bounds for the arithmetic circuit size of the permanent. We confirm the conjecture in a probabilistic sense by proving that if the coefficients involved in the description of f are independent standard Gaussian random variables, then the expected number of real zeros of f is .

Theta Functions and Brownian Motion

A theta function for an arbitrary connected and simply connected compact simple Lie group is defined as an infinite determinant that is naturally related to the transformation of a family of independent Gaussian random variables associated with a pinned Brownian motion in the Lie group. From this definition of a theta function, the equality of the product and the sum expressions for a theta function is obtained. This equality for an arbitrary connected and simply connected compact simple Lie group is known as a Macdonald identity which generalizes the Jacobi triple product for the elliptic theta function associated with su(2).

An Optimal Polynomial Approximation of Brownian Motion

In this paper, we will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. The coefficients that are obtained from the expansion of Brownian motion in this polynomial basis are independent Gaussian random variables. Therefore it is practical (requires $N$ independent Gaussian coefficients) to generate an approximate sample path of Brownian motion that respects integration of polynomials with degree less than $N$. Moreover, since these orthogonal polynomials appear naturally as eigenfunctions of an integral operator defined by the Brownian bridge covariance function, the proposed approximation is optimal in a certain weighted $L^{2}(\mathbb{P})$ sense. In addition, discretizing Brownian paths as piecewise parabolas gives a locally higher order numerical method for stochastic differential equations (SDEs) when compared to the standard piecewise linear approach. We shall demonstrate these ideas by simulating Inhomogeneous Geometric Brownian Motion (IGBM). This numerical example will also illustrate the deficiencies of the piecewise parabola approximation when compared to a new version of the asymptotically efficient log-ODE (or Castell-Gaines) method.

Cross Term Decay in Multiplicative Processors

Multiplicative processors combine the beamformed outputs of undersampled subarrays to estimate the spatial power spectrum. While the multiplicative processor requires fewer sensors to achieve the same resolution as a conventional linear processor, it also produces cross terms that can degrade the spectral estimate. Cross terms are false peaks formed when a signal passing through the grating lobe of one subarray interacts with a different signal passing through the other subarray. Snapshot averaging reduces cross terms when the signals are uncorrelated. This letter quantifies the number of snapshots required for effective cross term mitigation, accounting for the use of subarray tapers to control sidelobe levels. To facilitate the analysis, the letter derives closed form expressions for the probability density function and cumulative distribution function of the sum of products of independent complex Gaussian random variables. The analysis demonstrates that cross terms decay at a rate of 5 dB per decade of snapshots averaged. While subarray tapering reduces sidelobe leakage, it also increases the number of snapshots required for cross term mitigation.

Compound zero-truncated Poisson normal distribution and its applications

Here, we first propose three-parameter model and call it as the compound zero-truncated Poisson normal (ZTP-N) distribution. The model is based on the random sum of N independent Gaussian random variables, where N is a zero truncated Poisson random variable. The proposed ZTP-N distribution is a very flexible probability distribution function. The probability density function can take variety of shapes. It can be both positively and negatively skewed, moreover, normal distribution can be obtained as a special case. It can be unimodal, bimodal as well as multimodal also. It has three parameters. An efficient EM type algorithm has been proposed to compute the maximum likelihood estimators of the unknown parameters. We further propose a four-parameter bivariate distribution with continuous and discrete marginals, and discuss estimation of unknown parameters based on the proposed EM type algorithm. Some simulation experiments have been performed to see the effectiveness of the proposed EM type algorithm, and one real data set has been analyzed for illustrative purposes.