Outlier Eigenvalues for Deformed I.I.D. Random Matrices

We consider a square random matrix of size N of the form A + Y where A is deterministic and Y has i.i.d. entries with variance 1/N. Under mild assumptions, as N grows the empirical distribution of the eigenvalues of A + Y converges weakly to a limit probability measure β on the complex plane. This work is devoted to the study of the outlier eigenvalues, i.e., eigenvalues in the complement of the support of β. Even in the simplest cases, a variety of interesting phenomena can occur. As in earlier works, we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of Y or the Jordan decomposition of A. We also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a Gaussian analytic function. © 2016 Wiley Periodicals, Inc.     

Limit theorems for bivariate Appell polynomials. Part I: Central limit theorems

Summary. Consider the stationary linear process , , where is an i.i.d. finite variance sequence. The spectral density of may diverge at the origin (long-range dependence) or at any other frequency. Consider now the quadratic form , where denotes a non-linear function (Appell polynomial). We provide general conditions on the kernels and for to converge to a Gaussian distribution. We show that this convergence holds if and are not too badly behaved. However, the good behavior of one kernel may compensate for the bad behavior of the other. The conditions are formulated in the spectral domain. Received: 28 February 1996 / In revised form: 10 July 1996     

Fractional normal inverse Gaussian diffusion

A fractional normal inverse Gaussian (FNIG) process is a fractional Brownian motion subordinated to an inverse Gaussian process. This paper shows how the FNIG process emerges naturally as the limit of a random walk with correlated jumps separated by i.i.d. waiting times. Similarly, we show that the NIG process, a Brownian motion subordinated to an inverse Gaussian process, is the limit of a random walk with uncorrelated jumps separated by i.i.d. waiting times. The FNIG process is also derived as the limit of a fractional ARIMA processes. Finally, the NIG densities are shown to solve the relativistic diffusion equation from statistical physics.     

Asymptotic description of stochastic neural networks. I. Existence of a large deviation principle

We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. The dynamics of the neurons is described by a set of stochastic differential equations in discrete time. The neurons interact through the synaptic weights that are Gaussian correlated random variables. We describe the asymptotic law of the network when the number of neurons goes to infinity. Unlike previous works which made the biologically unrealistic assumption that the weights were i.i.d. random variables, we assume that they are correlated. We introduce the process-level empirical measure of the trajectories of the solutions into the equations of the finite network of neurons and the averaged law (with respect to the synaptic weights) of the trajectories of the solutions into the equations of the network of neurons. The result ( Theorem 3.1 below) is that the image law through the empirical measure satisfies a large deviation principle with a good rate function. We provide an analytical expression of this rate function in terms of the spectral representation of certain Gaussian processes.     

I.i.d. representations for the bivariate product limit estimators and the bootstrap versions

Let F(s, t) = P(X > s, Y > t) be the bivariate survival function which is subject to random censoring. Let be the bivariate product limit estimator (PL-estimator) by Campbell and Földes (1982, Proceedings International Colloquium on Non-parametric Statistical Inference, Budapest 1980, North-Holland, Amsterdam). In this paper, it was shown that

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, where {ζ_i(s, t)} is i.i.d. mean zero process and R_n(s, t) is of the order O((n⁻¹log n)^3/4) a.s. uniformly on compact sets. Weak convergence of the process {n⁻¹ Σ_{i = 1}ⁿ ζ_i(s, t)} to a two-dimensional-time Gaussian process is shown. The covariance structure of the limiting Gaussian process is also given. Corresponding results are also derived for the bootstrap estimators. The result can be extended to the multivariate cases and are extensions of the univariate case of Lo and Singh (1986, Probab. Theory Relat. Fields, 71, 455–465). The estimator is also modified so that the modified estimator is closer to the true survival function than in supnorm.     

The periodogram of an i.i.d. sequence

Periodogram ordinates of a Gaussian white-noise computed at Fourier frequencies are well known to form an i.i.d. sequence. This is no longer true in the non-Gaussian case. In this paper, we develop a full theory for weighted sums of non-linear functionals of the periodogram of an i.i.d. sequence. We prove that these sums are asymptotically Gaussian under conditions very close to those which are sufficient in the Gaussian case, and that the asymptotic variance differs from the Gaussian case by a term proportional to the fourth cumulant of the white noise. An important consequence is a functional central limit theorem for the spectral empirical measure. The technique used to obtain these results is based on the theory of Edgeworth expansions for triangular arrays.     

Linear relations in time series models. I

A multiple time series is defined as the sum of an autoregressive process on a line and independent Gaussian white noise on a hyperplane that goes through the origin and intersects the line at a single point. This process is a multiple autoregressive time series in which the regression matrices satisfy suitable conditions. It is shown that the maximum likelihood estimates of the line and the autoregression coefficients can be obtained as the values that minimize a given function, and that the remaining maximum likelihood estimates can be computed as simple functions of the first ones. It is also shown that the maximum likelihood estimates are equivariant with respect to the group of bijective linear transformations.     

The empirical distribution function and partial sum process of residuals from a stationary arch with drift process

The weak convergence of the empirical process and partial sum process of the residuals from a stationary ARCH-M model is studied. It is obtained for and consistent estimate of the ARCH-M parameters. We find that the limiting Gaussian processes are no longer distribution free and hence residuals cannot be treated as i.i.d. In fact the limiting Gaussian process for the empirical process is a standard Brownian bridge plus an additional term, while the one for partial sum process is a standard Brownian motion plus an additional term. In the special case of a standard ARCH process, that is an ARCH process with no drift, the additional term disappears. We also study a sub-sampling technique which yields the limiting Gaussian processes for the empirical process and partial sum process as a standard Brownian bridge and a standard Brownian motion respectively.     

Local Limit Theorem for the Supremum of an Empirical Process for I.I.D. Random Variables

In this paper, we establish the convergence in total-variation norm of the law of the supremum of an empirical process constructed from a sequence of i.i.d. random variables to the law of the supremum of a (generalized) Brownian bridge. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 457–478, October–December, 2005.     

Simulation accuracy of stationary Gaussian stochastic processes inL 2(0,T)

Models of stationary Gaussian stochastic processes with discrete and continuous spectra are constructed. Simulation of stationary Gaussian processes with a continuous spectrum is considered for the following cases: when the covariance function of the stochastic process is expandable in a Fourier series with positive coefficients; when the spectrum of the stationary Gaussian stochastic process is concentrated on the interval [0, ]; and in the general case. The stationary Gaussian process is simulated with prescribed reliability and accuracy in L²(0, T).Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 108–115, 1991.     

Fractal measures and polynomial sampling: I.F.S.–Gaussian integration

Measures generated by Iterated Function Systems can be used in place of atomic measures in Gaussian integration. A stable algorithm for the numerical solution of the related approximation problem – an inverse problem in fractal construction – is proposed. Dedicated to Walter Gautschi.     

关于独立同分布正态随机变量部分和尾概率的注(英文)

设{X,Xn,n≥1}是独立同分布正态随机变量序列,EX=0且EX2=σ2>0,Sn=sum (Xk) form k=1 to n,λ(ε) =sum form (P(|Sn|≥ nε)) form n=1 to ∞.在本文中,我们证明了存在正常数C1和C2,使得对足够小的ε>0,成立下列不等式C1ε3 ≤ε2λ(ε)-σ2+ε2 /2 ≤ C2ε3.     

Probability of a stationary stochastic process staying in a strip

We consider the distribution of the duration of a stochastic process staying in a strip. An example is given of the calculation of the distribution for a Gaussian cosine process.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 128–130, 1988.     

An asymptotic decomposition for multivariate distribution-free tests of independence

In the multivariate case, the empirical dependence function, defined as the empirical distribution function with reduced uniform margins on the unit interval, can be shown for an i.i.d. sequence to converge weakly in an asymptotic way to a limiting Gaussian process. The main result of this paper is that this limiting process can be canonically separated into a finite set of independent Gaussian processes, enabling one to test the existence of dependence relationships within each subset of coordinates independently (in an asymptotic way) of what occurs in the other subsets. As an application we derive the Karhunen-Loeve expansions of the corresponding processes and give the limiting distribution of the multivariate Cramer-Von Mises test of independence, generalizing results of Blum, Kiefer, Rosenblatt, and Dugué. Other extensions are mentioned, including a generalization of Kendall's τ.     

Asymptotic properties of correlation bounds in functional spaces. I.

A bound is constructed for the correlation function of a uniform Gaussian random field in the scheme of series with respect to the many samples. Exact properties are established for the bound. It is proved that it is strongly consistent and asymptotically normal in the Hilbert space of functions which are square integrable on R^m with same weight function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 179–187, February, 1991.     

Approximation of Gaussian measures generated by stationary processes. I

We consider local approximation of Gaussian measures generated by stationary processes, which are smooth functions of a parameter.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 79, pp. 44–66, 1978.     

Limiting distributions of the solutions of the many-dimensional Bürgers equation with random initial data. I

We find Gaussian limiting distributions of the solutions of the many-dimensional Bürgers equation with the initial condition given by a homogeneous isotropic Gaussian random field with strong dependence.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 870–877, July, 1994.     

Strong approximations for partial sums of i.i.d.B-valued r.v.'s in the domain of attraction of a Gaussian law

Summary We obtain a strong approximation theorem for partial sums of i.i.d.d-dimensional r.v.'s with possibly infinite second moments. Using this result, we can extend Philipp's strong invariance principle for partial sums of i.i.d.B-valued r.v.'s satisfying the central limit theorem toB-valued r.v.'s which are only in the domain of attraction of a Gaussian law. This new strong invariance principle implies a compact as well as a functional law of the iterated logarithm which improve some recent results of Kuelbs (1985).     

Zariski closure and the dimension of the Gaussian low of the product of random matrices. I

Summary We prove the Central Limit Theorem for products of i.i.d. random matrices. The main aim is to find the dimension of the corresponding Gaussian law. It turns out that ifG is the Zariski closure of a group generated by the support of the distribution of our matrices, and ifG is semi-simple, then the dimension of the Gaussian law is equal to the dimension of the diagonal part of Cartan decomposition ofG.In this article we present a detailed exposition of results announced in [GGu]. For reasons explained in the introduction, this part is devoted to the case ofSL(m, ) group. The general semi-simple Lie group will be considered in the second part of the work.The central limit theorem for products of independent random matrices is our main topic, and the results obtained complete to a large extent the general picture of the subject.The proofs rely on methods from two theories. One is the theory of asymptotic behaviour of products of random matrices itself. As usual, the existence of distinct Lyapunov exponents is the most important fact here. The other is the theory of algebraic groups. We want to point out that algebraic language and methods play a very important role in this paper.In fact, this mixture of methods has already been used for the study of Lyapunov exponents in [GM1, GM2, GR3]. We believe that it is impossible to avoid the algebraic approach if one aims to obtain complete and effective answers to natural problems arising in the theory of products of random matrices.In order also to present the general picture of the subject we describe several results which are well known. Some of these can be proven for stationary sequences of matrices, others are true also for infinite dimensional operators (see e.g. [BL, O, GM2, L, R]). But our main concern is with independent matrices, in which case very precise and constructive statements can be obtained.     

Convergence conditions for variance estimator of Gaussian stationary stochastic processes in Orlicz spaces

Conditions are derived ensuring convergence of the variance estimator of a Gaussian stationary stochastic process in Orlicz space norm. Confidence intervals are constructed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 113–116, 1992.