Stable and Semistable Hemigroups: Domains of Attraction and Self-Decomposability

A hemigroup of probability measures builds the set of distributions of the increments of an independent increment process. Decomposability properties of the hemigroup lead to stable respectively semistable hemigroups and enable us to show that such hemigroups appear as limits of certain functional limit theorems for operator-normed independent (non-identically) distributed random vectors. Regular variation properties of the norming operators show that the functional limit theorems are closely related to limits of infinitesimal triangular arrays of independent random vectors, i.e., to operator-self-decomposable respectively operator-semi-self-decomposable laws.     

Multidimensional large deviation local limit theorems

An earlier paper by the author ([4], 97–114) established large deviation local limit theorems for arbitrary sequences of real valued random variables. This work showed clearly the connection between the Cramér series and large deviation rates. In this article we present large deviation local limit theorems for arbitrary multidimensional random variables based solely on conditions imposed on their moment generating functions. These results generalize the theorems of [12], 100–106) for sums of independent and identically distributed random vectors.     

Limit theorems for functionals of Gaussian vectors

Operator self-similar processes, as an extension of self-similar processes, have been studied extensively. In this work, we study limit theorems for functionals of Gaussian vectors. Under some conditions, we determine that the limit of partial sums of functionals of a stationary Gaussian sequence of random vectors is an operator self-similar process.     

Structure of exchangeable infinitely divisible sequences of poisson random vectors

De Finetti's Theorem reveals a simple explicit structure for an infinite exchangeable sequence of zero-one random variables. Although more general results are known, simple explicit results might be expected in particular settings. In this paper such results are obtained for exchangeable sequences of infinitely divisible Poisson random variables and random vectors. The methods employed are elementary, except in that they involve appeal to moment theorems.     

On a Class of Covariance Operators

This paper is the continuation of [1] in which complex symmetries of distributions and their covariance operators are investigated. Here we also study the most general quaternion symmetries of random vectors. Complete classification theorems on these symmetries are proved in terms of covariance operator spectra.     

Toward the History of the Saint Petersburg School of Probability and Statistics. I. Limit Theorems for Sums of Independent Random Variables

This is the first in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg school of probability and statistics in the period from 1947 to 2017. It is devoted to limit theorems for sums of independent random variables—a traditional subject for St. Petersburg. It refers to the classical limit theorems: the law of large numbers, the central limit theorem, and the law of the iterated logarithm, as well as important relevant problems formulated in the second half of the twentieth century. The latter include the approximation of the distributions of sums of independent variables by infinitely divisible distributions, estimation of the accuracy of strong Gaussian approximation of such sums, and the limit theorems on the weak almost sure convergence of empirical measures generated by sequences of sums of independent random variables and vectors.     

On large deviations for sums of random vectors in Rk

The paper deals with limit theorems for probabilities of large deviations for sums of independent identically distributed random vectors. We give more detailed bounds for the remainder in von Bahr's limit theorem. New asymptotic formulas for probabilities of large deviations on the outside of balls are established.     

A Multivariate CLT for Decomposable Random Vectors with Finite Second Moments

Stein's method is used to derive a CLT for dependent random vectors possessing the dependence structure from Barbour et al. J. Combin. Theory Ser. B 47, 125–145, but under the assumption of second moments only. This allows us to derive Lindeberg–Feller type theorems for sums of random vectors with certain dependence structures. We apply the main theorem to the study of three problems: local dependence, random graph degree statistics and finite population statistics. In particular, we consider U-statistics of independent observations as well as of observations drawn without replacement.     

α-Stable limit theorems for sums of dependent random vectors

Several α-stable limit theorems for sums of dependent random vectors are proved via point processes theory; p-mixing, m-dependence, and the type of mixing treated within the extreme value theory are considered.     

Random capacities and their distributions

Summary We formalize the notion of an increasing and outer continuous random process, indexed by a class of compact sets, that maps the empty set on zero. Existence and convergence theorems for distributions of such processes are proved. These results generalize or are similar to those known in the special cases of random measures, random (closed) sets and random (upper) semicontinuous functions. For the latter processes infinite divisibility under the maximum is introduced and characterized. Our result generalizes known characterizations of infinite divisibility for random sets and max-infinite divisibility for random vectors. Also discussed is the convergence in distribution of the row-vise maxima of a null-array of random semicontinuous functions.Research supported by the Swedish Natural Science Research Council     

Stochastic stable population growth

Classical demographic theory purports that the age structure of a population eventually stabilizes. Although the population may continue to grow, once equilibrium is reached, the proportions of people in different age categories do not change. Stochastic analogues can be proven if vital rates fluctuate according to a stationary stochastic process. The action of random matrix products on random vectors is studied. This permits the application of Hilbert's projective metric and leads to considerable simplification of the ergodic and central limit theory of population growth. Appropriate theorems and their proofs are presented.     

Rapid Growth Paths in Multivalued Dynamical Systems Generated by Homogeneous Convex Stochastic Operators

The paper examines dynamical systems generated by convex homogeneous multivalued operators in spaces of random vectors. The primary goal is to investigate the growth rates of random trajectories of these dynamical systems. Existence and characterization theorems for rapid trajectories, growing faster in a certain sense than others, are obtained.     

Triangular array limits for continuous time random walks

A continuous time random walk (CTRW) is a random walk subordinated to a renewal process, used in physics to model anomalous diffusion. Transition densities of CTRW scaling limits solve fractional diffusion equations. This paper develops more general limit theorems, based on triangular arrays, for sequences of CTRW processes. The array elements consist of random vectors that incorporate both the random walk jump variable and the waiting time preceding that jump. The CTRW limit process consists of a vector-valued Lévy process whose time parameter is replaced by the hitting time process of a real-valued nondecreasing Lévy process (subordinator). We provide a formula for the distribution of the CTRW limit process and show that their densities solve abstract space–time diffusion equations. Applications to finance are discussed, and a density formula for the hitting time of any strictly increasing subordinator is developed.     

On fuzzy random multiobjective quadratic programming

In this paper, a multiobjective quadratic programming problem having fuzzy random coefficients matrix in the objective and constraints and the decision vector are fuzzy pseudorandom variables is considered. First, we show that the efficient solutions of fuzzy quadratic multiobjective programming problems are resolved into series-optimal-solutions of relative scalar fuzzy quadratic programming. Some theorems are proved to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients, having decision vectors as fuzzy variables. At the end, numerical examples are illustrated in the support of the obtained results.     

随机积分和微分方程在弱拓扑下解的存在性和比较结果*

在本文中,我们对非线性随机Volterra积分方程在Banach空间的弱拓扑下的随机解证明了几个存在定理.然后作为应用,我们得到了随机微分方程的弱随机解的存在定理.还得到了这些随机方程的极值随机解的存在性和随机比较定理.我们的定理改进和推广了[4,5,10,11,12]中的相应结果.     

Extreme Value Asymptotics for Multivariate Renewal Processes

For a sequence of partial sums ofd-dimensional independent identically distributed random vectors a corresponding multivariate renewal process is defined componentwise. Via strong invariance together with an extreme value limit theorem for Rayleigh processes, a number of weak asymptotic results are established for thed-dimensional renewal process. Similar theorems for the estimated version of this process are also derived. These results are suggested to serve as simultaneous asymptotic testing devices for detecting changes in the multivariate setting.     

Limit theorems for random transformations and processes in random environments

I derive general relativized central limit theorems and laws of iterated logarithm for random transformations both via certain mixing assumptions and via the martingale differences approach. The results are applied to Markov chains in random environments, random subshifts of finite type, and random expanding in average transformations where I show that the conditions of the general theorems are satisfied and so the corresponding (fiberwise) central limit theorems and laws of iterated logarithm hold true in these cases. I consider also a continuous time version of such limit theorems for random suspensions which are continuous time random dynamical systems.

     

The existence theorems of the random solutions for random Hammerstein type nonlinear integral equations

This paper deals with the existence theorems of random solutions of random Hammerstein type nonlinear integral equations. These theorems are proved by using the random fixed-point theorems of cone expansion and compression of random operator discussed by Li and Sheng [1].     

Singular vectors under random perturbation

Computing the first few singular vectors of a large matrix is a problem that frequently comes up in statistics and numerical analysis. Given the presence of noise, an exact calculation is hard to achieve, and the following problem is of importance: How much does a small perturbation to the matrix change the singular vectors? Answering this question, classical theorems, such as those of Davis‐Kahan and Wedin, give tight estimates for the worst‐case scenario. In this paper, we show that if the perturbation (noise) is random and our matrix has low rank, then better estimates can be obtained. Our method relies on high dimensional geometry and is different from those used in earlier papers. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

The existence theorems of the random solutions for random Hammerstein equation with random kernel (II)

In this paper, we use the random fixed theorems of cone expansion and compression of random operator to obtain the existence theorems of random solutions for random Hammerstein integral equation of polynomial type with random kernel.