Showing papers in "Journal of Theoretical Probability in 2012"

How Close is the Sample Covariance Matrix to the Actual Covariance Matrix

Abstract: Given a probability distribution in ℝ n with general (nonwhite) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent points. What is the optimal sample size N=N(n) that guarantees estimation with a fixed accuracy in the operator norm? Suppose that the distribution is supported in a centered Euclidean ball of radius $O(\sqrt{n})$ . We conjecture that the optimal sample size is N=O(n) for all distributions with finite fourth moment, and we prove this up to an iterated logarithmic factor. This problem is motivated by the optimal theorem of Rudelson (J. Funct. Anal. 164:60–72, 1999), which states that N=O(nlog n) for distributions with finite second moment, and a recent result of Adamczak et al. (J. Am. Math. Soc. 234:535–561, 2010), which guarantees that N=O(n) for subexponential distributions.

Limit Theorems for Weakly Subcritical Branching Processes in Random Environment

Abstract: For a branching process in random environment, it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, “supercritical.” This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on nonextinction. Also a functional limit theorem is proved, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.

A Particle Migrating Randomly on a Sphere

Abstract: Consider a particle moving on the surface of the unit sphere in R 3 and heading towards a specific destination with a constant average speed, but subject to random deviations.

SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations

Abstract: It is known that the transition probabilities of a solution to a classical Ito stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coefficients determined by the corresponding SDE. Time-fractional Kolmogorov-type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distributed order differential equations, the associated class of SDEs can be described within the framework of SDEs driven by semimartingales. These semimartingales are time-changed Levy processes where the independent time-change is given respectively by the inverse of a single or mixture of independent stable subordinators. Examples are provided, including a fractional analogue of the Feynman–Kac formula.

Transition Density Estimates for a Class of Lévy and Lévy-Type Processes

Abstract: We show on- and off-diagonal upper estimates for the transition densities of symmetric Levy and Levy-type processes. To get the on-diagonal estimates, we prove a Nash-type inequality for the related Dirichlet form. For the off-diagonal estimates, we assume that the characteristic function of a Levy(-type) process is analytic, which allows us to apply the complex analysis technique.

Finite Variation of Fractional Lévy Processes

Abstract: Various characterizations for fractional Levy processes to be of finite variation are obtained, one of which is in terms of the characteristic triplet of the driving Levy process, while others are in terms of differentiability properties of the sample paths. A zero-one law and a formula for the expected total variation are also given.

Spectra of Large Random Trees

Abstract: We analyze the eigenvalues of the adjacency matrices of a wide variety of random trees. Using general, broadly applicable arguments based on the interlacing inequalities for the eigenvalues of a principal submatrix of a Hermitian matrix and a suitable notion of local weak convergence for an ensemble of random trees that we call probability fringe convergence, we show that the empirical spectral distributions for many random tree models converge to a deterministic (model-dependent) limit as the number of vertices goes to infinity.

Exponents, Symmetry Groups and Classification of Operator Fractional Brownian Motions

Abstract: Operator fractional Brownian motions (OFBMs) are zero mean, operator self-similar (o.s.s.) Gaussian processes with stationary increments. They generalize univariate fractional Brownian motions to the multivariate context. It is well-known that the so-called symmetry group of an o.s.s. process is conjugate to subgroups of the orthogonal group. Moreover, by a celebrated result of Hudson and Mason, the set of all exponents of an operator self-similar process can be related to the tangent space of its symmetry group.

A Gaussian Inequality for Expected Absolute Products

Abstract: We prove the inequality that \({\mathbb{E}}|X_{1}X_{2}\cdots X_{n}|\leq \sqrt{\mathrm{per}(\varSigma )}\), for any centered Gaussian random variables X1,…,Xn with the covariance matrix Σ, followed by several applications and examples We also discuss a conjecture on the lower bound of the expectation

Malliavin Calculus for Fractional Delay Equations

Abstract: In this article, we give some existence and smoothness results for the law of the solution to a stochastic heat equation driven by a finite dimensional fractional Brownian motion with Hurst parameter $H>1/2$. Our results rely on recent tools of Young integration for convolutional integrals combined with stochastic analysis methods for the study of laws of random variables defined on a Wiener space.

Central Limit Theorem Started at a Point for Stationary Processes and Additive Functionals of Reversible Markov Chains

Abstract: In this paper we study the almost sure central limit theorem started at a point for additive functionals of a stationary and ergodic Markov chain via a martingale approximation in the almost sure sense. Some of the results provide sufficient conditions for general stationary sequences. We use these results to study the quenched CLT for additive functionals of reversible Markov chains.

Free Infinite Divisibility of Free Multiplicative Mixtures of the Wigner Distribution

Abstract: Let I * and I ⊞ be the classes of all classical infinitely divisible distributions and free infinitely divisible distributions, respectively, and let Λ be the Bercovici–Pata bijection between I * and I ⊞ . The class type W of symmetric distributions in I ⊞ that can be represented as free multiplicative convolutions of the Wigner distribution is studied. A characterization of this class under the condition that the mixing distribution is 2-divisible with respect to free multiplicative convolution is given. A correspondence between symmetric distributions in I ⊞ and the free counterpart under Λ of the positive distributions in I * is established. It is shown that the class type W does not include all symmetric distributions in I ⊞ and that it does not coincide with the image under Λ of the mixtures of the Gaussian distribution in I *. Similar results for free multiplicative convolutions with the symmetric arcsine measure are obtained. Several well-known and new concrete examples are presented.

Hitting Time Distributions for Denumerable Birth and Death Processes

Abstract: For an ergodic continuous-time birth and death process on the nonnegative integers, a well-known theorem states that the hitting time T 0,n starting from state 0 to state n has the same distribution as the sum of n independent exponential random variables. Firstly, we generalize this theorem to an absorbing birth and death process (say, with state −1 absorbing) to derive the distribution of T 0,n . We then give explicit formulas for Laplace transforms of hitting times between any two states for an ergodic or absorbing birth and death process. Secondly, these results are all extended to birth and death processes on the nonnegative integers with ∞ an exit, entrance, or regular boundary. Finally, we apply these formulas to fastest strong stationary times for strongly ergodic birth and death processes.

Approximation of Projections of Random Vectors

Abstract: Let X be a d-dimensional random vector and Xθ its projection onto the span of a set of orthonormal vectors {θ1,…,θk} Conditions on the distribution of X are given such that if θ is chosen according to Haar measure on the Stiefel manifold, the bounded-Lipschitz distance from Xθ to a Gaussian distribution is concentrated at its expectation; furthermore, an explicit bound is given for the expected distance, in terms of d, k, and the distribution of X, allowing consideration not just of fixed k but of k growing with d The results are applied in the setting of projection pursuit, showing that most k-dimensional projections of n data points in ℝd are close to Gaussian, when n and d are large and k=clog (d) for a small constant c

General Moments of the Inverse Real Wishart Distribution and Orthogonal Weingarten Functions

Abstract: We study a random positive definite symmetric matrix distributed according to a real Wishart distribution. We compute general moments of the random matrix and of its inverse explicitly. To do so, we employ the orthogonal Weingarten function, which was recently introduced in the study of Haar-distributed orthogonal matrices. As applications, we give formulas for moments of traces of a Wishart matrix and its inverse.

Sublinear Variance for Directed Last-Passage Percolation

Abstract: A range of first-passage percolation type models are believed to demonstrate the related properties of sublinear variance and superdiffusivity. We show that directed last-passage percolation with Gaussian vertex weights has a sublinear variance property. We also consider other vertex weight distributions.

A Stochastic Stefan Problem

Abstract: We consider a stochastic perturbation of the Stefan problem. The noise is Brownian in time and smoothly correlated in space. We prove existence and uniqueness and characterize the domain of existence.

Asymptotic Estimates of the Distribution of Brownian Hitting Time of a Disc

Abstract: The distribution of the first hitting time of a disc for the standard two-dimensional Brownian motion is computed. By investigating the inversion integral of its Laplace transform we give fairly detailed asymptotic estimates of its density valid uniformly with respect to the point from which the Brownian motion starts.

Asymptotic Properties of the Markov Branching Process with Immigration

Abstract: We consider the Markov branching process with immigration allowing the possibility of infinite numbers of offspring and/or immigrants. Our focus is on the construction and uniqueness of the minimal transition function and on its asymptotic behavior. Conditional limit theorems for the population size are given in cases for which the transition function is dishonest.

Second order Subexponential Distributions with Finite Mean and Their Applications to Subordinated Distributions

Abstract: NSFC [10926043]; Fundamental Research Funds for the Central Universities in China [2010121005]

A Generalized Comparison Theorem for BSDEs and Its Applications

Abstract: This paper establishes a generalized comparison theorem for one-dimensional backward stochastic differential equations (BSDEs) whose generators are uniformly continuous in z and satisfy a kind of weakly monotonic condition in y. As applications, two new existence and uniqueness theorems for solutions of BSDEs are obtained. In the one-dimensional setting, these results generalize some corresponding results in Pardoux and Peng (Syst. Control Lett. 14:55–61, 1990), Mao (Stoch. Process. Their Appl. 58:281–292, 1995), El Karoui et al. (Math. Finance 7:1–72, 1997), Pardoux (Nonlinear Analysis, Differential Equations and Control, Montreal, QC, 1998, Kluwer Academic, Dordrecht, 1999), Cao and Yan (Adv. Math. 28(4):304–308, 1999), Briand and Hu (Probab. Theory Relat. Fields 136(4):604–618, 2006), and Jia (C. R. Acad. Sci. Paris, Ser. I 346:439–444, 2008).

Lévy’s Zero–One Law in Game-Theoretic Probability

Abstract: We prove a nonstochastic version of Levy’s zero–one law and deduce several corollaries from it, including nonstochastic versions of Kolmogorov’s zero–one law and the ergodicity of Bernoulli shifts. Our secondary goal is to explore the basic definitions of game-theoretic probability theory, with Levy’s zero–one law serving a useful role.

Distribution of Eigenvalues of Highly Palindromic Toeplitz Matrices

Abstract: Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous work (Bryc et al., Ann. Probab. 34(1):1–38, 2006; Hammond and Miller, J. Theor. Probab. 18(3):537–566, 2005) showed that the spectral measures (the density of normalized eigenvalues) converge almost surely to a universal distribution almost that of the Gaussian, independent of p. The deficit from the Gaussian distribution is due to obstructions to solutions of Diophantine equations and can be removed (see Massey et al., J. Theor. Probab. 20(3):637–662, 2007) by making the first row palindromic. In this paper we study the case where there is more than one palindrome in the first row of real symmetric Toeplitz matrices. Using the method of moments and an analysis of the resulting Diophantine equations, we show that the spectral measures converge almost surely to a universal distribution. Assuming a conjecture on the resulting Diophantine sums (which is supported by numerics and some theoretical arguments), we prove that the limiting distribution has a fatter tail than any previously seen limiting spectral measure.

Central Limit Theorems for Uniform Model Random Polygons

Abstract: We show how a central limit theorem for Poisson model random polygons implies a central limit theorem for uniform model random polygons. To prove this implication, it suffices to show that in the two models, the variables in question have asymptotically the same expectation and variance. We use integral geometric expressions for these expectations and variances to reduce the desired estimates to the convergence \((1+\frac{\alpha}{n})^{n}\to e^{\alpha}\) as n→∞.

Occupation Time Fluctuations of Weakly Degenerate Branching Systems

Abstract: We establish limit theorems for rescaled occupation time fluctuations of a sequence of branching particle systems in ℝ d with anisotropic space motion and weakly degenerate splitting ability. In the case of large dimensions, our limit processes lead to a new class of operator-scaling Gaussian random fields with nonstationary increments. In the intermediate and critical dimensions, the limit processes have spatial structures analogous to (but more complicated than) those arising from the critical branching particle system without degeneration considered by Bojdecki et al. (Stoch. Process. Appl. 116:1–18 and 19–35, 2006). Due to the weakly degenerate branching ability, temporal structures of the limit processes in all three cases are different from those obtained by Bojdecki et al. (Stoch. Process. Appl. 116:1–18 and 19–35, 2006).

A Universality Property of Gaussian Analytic Functions

Abstract: 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.

Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients

Abstract: In this paper we develop a new weak convergence and compact embedding method to study the existence and uniqueness of the $L_{\rho}^{2p}({\mathbb{R}^{d}};{\mathbb{R}^{1}})\times L_{\rho}^{2}({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of backward stochastic differential equations with p-growth coefficients. Then we establish the probabilistic representation of the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.

On Regularity Property of Retarded Ornstein-Uhlenbeck Processes in Hilbert Spaces

Abstract: In this work, some regularity properties of mild solutions for a class of stochastic linear functional differential equations driven by infinite-dimensional Wiener processes are considered. In terms of retarded fundamental solutions, we introduce a class of stochastic convolutions which naturally arise in the solutions and investigate their Yosida approximants. By means of the retarded fundamental solutions, we find conditions under which each mild solution permits a continuous modification. With the aid of Yosida approximation, we study two kinds of regularity properties, temporal and spatial ones, for the retarded solution processes. By employing a factorization method, we establish a retarded version of the Burkholder–Davis–Gundy inequality for stochastic convolutions.

Precise Large Deviations for Long-Tailed Distributions

Abstract: In this paper, we investigate the precise large deviations for sums of independent identically distributed random variables with heavy-tailed distributions. We prove asymptotic relations for non-random sums and for random sums of random variables with long-tailed distributions. We apply the results on two useful counting processes, namely, renewal and compound-renewal processes.

Matrix Measures, Random Moments, and Gaussian Ensembles

Abstract: We consider the moment space $\mathcal{M}_{n}$ corresponding to p×p real or complex matrix measures defined on the interval [0,1]. The asymptotic properties of the first k components of a uniformly distributed vector $(S_{1,n}, \dots , S_{n,n})^{*} \sim\mathcal{U} (\mathcal{M}_{n})$ are studied as n→∞. In particular, it is shown that an appropriately centered and standardized version of the vector (S 1,n ,…,S k,n )∗ converges weakly to a vector of k independent p×p Gaussian ensembles. For the proof of our results, we use some new relations between ordinary moments and canonical moments of matrix measures which are of their own interest. In particular, it is shown that the first k canonical moments corresponding to the uniform distribution on the real or complex moment space $\mathcal{M}_{n}$ are independent multivariate Beta-distributed random variables and that each of these random variables converges in distribution (as the parameters converge to infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary ensemble, respectively.