CLT for linear random fields with stationary martingale-difference innovation

In [V. Paulauskas, On Beveridge–Nelson decomposition and limit theorems for linear random fields, J. Multivariate Anal., 101:621–639, 2010], limit theorems for linear random fields generated by independent identically distributed innovations were proved. In this paper, we present the central limit theorem for linear random fields with martingale-differences innovations satisfying the central limit theorem from [J. Dedecker, A central limit theorem for stationary random fields, Probab. Theory Relat. Fields, 110(3):397–426, 1998] and arranged in lexicographical order.     

On Beveridge-Nelson decomposition and limit theorems for linear random fields

We consider linear random fields and show how an analogue of the Beveridge-Nelson decomposition can be applied to prove limit theorems for sums of such fields.     

Veraverbeke’s theorem at large: on the maximum of some processes with negative drift and heavy tail innovations

Veraverbeke’s (Stoch Proc Appl 5:27–37, 1977) theorem relates the tail of the distribution of the supremum of a random walk with negative drift to the tail of the distribution of its increments, or equivalently, the probability that a centered random walk with heavy-tail increments hits a moving linear boundary. We study similar problems for more general processes. In particular, we derive an analogue of Veraverbeke’s theorem for fractional integrated ARMA models without prehistoric influence, when the innovations have regularly varying tails. Furthermore, we prove some limit theorems for the trajectory of the process, conditionally on a large maximum. Those results are obtained by using a general scheme of proof which we present in some detail and should be of value in other related problems.     

Normal approximation for linear stochastic processes and random fields in Hilbert space

The central limit theorem is proved for linear random fields defined on an integer-valued lattice of arbitrary dimension and taking values in Hilbert space. It is shown that the conditions in the central limit theorem are optimal. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 421–428, September, 2000.     

Two-dimensional Meixner Random Vectors of Class ${\mathcal{M}}_{L}$

The paper is divided into two parts. In the first part we lay down the foundation for defining the joint annihilation–preservation–creation decomposition of a finite family of not necessarily commutative random variables, and show that this decomposition is essentially unique. In the second part we show that any two, not necessarily commutative, random variables X and Y for which the vector space spanned by the identity and their annihilation, preservation, and creation operators equipped with the bracket given by the commutator forms a Lie algebra are equivalent up to an invertible linear transformation to two independent Meixner random variables with mixed preservation operators. In particular, if X and Y commute, then they are equivalent up to an invertible linear transformation to two independent classic Meixner random variables. To show this we start with a small technical condition called “non-degeneracy”.     

Limit distribution of the number of solutions of a system of random Boolean equations with a linear part

We prove two theorems on the Poisson limit distribution of the number of solutions of an a priori consistent system of nonlinear random Boolean equations with stochastically independent coefficients. In particular, we assume that this system contains a linear part. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1214–1226, September, 1998.     

Limit theorems for polynomial-type distributions

Distributions are found of independent nonnegative integer valued random variables under linear constraints. Limit theorems for these distributions are proved. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipolez, pp. 136–148, Perm, 1993.     

Rates of convergence in the CLT for linear random fields

In this paper, we study sums of linear random fields defined on the lattice Z ² with values in a Hilbert space. The rate of convergence of distributions of such sums to the Gaussian law is discussed, and mild sufficient conditions to obtain an approximation of order n ^−p are presented. This can be considered as a complement of a recent result of [A.N. Nazarova, Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space, Fundam. Prikl. Mat., 8:1091–1098, 2002 (in Russian)], where the logarithmic rate of convergence was stated, and as a generalization of the result of [D. Bosq, Erratum and complements to Berry–Esseen inequality for linear processes in Hilbert spaces, Stat. Probab. Lett., 70:171–174, 2004] for linear processes.     

On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria

In this paper we study the central limit theorem and its weak invariance principle for sums of non-adapted stationary sequences, under different normalizations. Our conditions involve the conditional expectation of the variables with respect to a given σ-algebra, as done in Gordin (Dokl. Akad. Nauk SSSR 188, 739–741, 1969) and Heyde (Z. Wahrsch. verw. Gebiete 30, 315–320, 1974). These conditions are well adapted to a large variety of examples, including linear processes with dependent innovations or regular functions of linear processes.     

Asymptotic of the heat kernel in general Benedicks domains

 Using a new inequality relating the heat kernel and the probability of survival, we prove asymptotic ratio limit theorems for the heat kernel (and survival probability) in general Benedicks domains. In particular, the dimension of the cone of positive harmonic measures with Dirichlet boundary conditions can be derived from the rate of convergence to zero of the heat kernel (or the survival probability). Received: 31 March 2002 / Revised version: 12 August 2002 / Published online: 19 December 2002 Mathematics Subject Classification (2000): 60J65, 31B05 Key words or phrases: Positive harmonic functions – Ratio limit theorems – Survival probability     

A Note on Asymptotic Normality of Kernel Estimation for Linear Random Fields on <Emphasis Type="Italic">Z</Emphasis><Superscript>2</Superscript>

This note considers the kernel estimation of a linear random field on Z ². Instead of imposing certain mixing conditions on the random fields, it is assumed that the weights of the innovations satisfy a summability property. By building a martingale decomposition based on a suitable filtration, asymptotic normality is proven for the kernel estimator of the marginal density of the random field. T.-L. Cheng’s research is supported in part by NSC 94-2118-M-018-001, Taiwan. Also, he is indebted to Department of Mathematics and Statistics, University of Calgary, for their hospitality during his visit. X. Lu’s research is supported in part by NSERC Discovery Grant of Canada.     

Asymptotic Normality of Kernel Type Density Estimators for Random Fields

Kernel type density estimators are studied for random fields. It is proved that the estimators are asymptotically normal if the set of locations of observations become more and more dense in an increasing sequence of domains. It turns out that in our setting the covariance structure of the limiting normal distribution can be a combination of those of the continuous parameter and the discrete parameter cases. The proof is based on a new central limit theorem for α-mixing random fields. Simulation results support our theorems. Final version 29 October 2004     

Generalized Knaster–Kuratowski–Mazurkiewicz Theorem Without Convex Hull

In this paper, a new notion of Knaster–Kuratowski–Mazurkiewicz mapping is introduced and a generalized Knaster–Kuratowski–Mazurkiewicz theorem is proved. As applications, some existence theorems of solutions for (vector) Ky Fan minimax inequality, Ky Fan section theorem, variational relation problems, n-person noncooperative game, and n-person noncooperative multiobjective game are obtained.     

The Mordel–Weil theorems for Drinfeld modules over finitely generated function fields

We generalize Poonen's analogue of Mordell–Weil theorems for Drinfeld modules over global function fields to the case of Drinfeld modules over finitely generated function fields. In addition, the A-characteristic of the function fields under our consideration can be arbitrary. Received: 23 April 2001     

Macroscaling Limit Theorems for Filtered Spatiotemporal Random Fields

This article addresses the problem of defining a general scaling setting in which Gaussian and non-Gaussian limit distributions of linear random fields can be obtained. The linear random fields considered are defined by the convolution of a Green kernel, satisfying suitable scaling conditions, with a non-linear transformation of a Gaussian centered homogeneous random field. The results derived cover the weak-dependence and strong-dependence cases for such Gaussian random fields. Extension to more general random initial conditions defined, for example, in terms of non-linear transformations of χ²-random fields, is also discussed. For an example, we consider the random fractional diffusion equation. The vectorial version of the limit theorems derived is also formulated, including the limit distribution of the parabolically rescaled solution to the Burgers equation in the cases of weakly and strongly dependent initial potentials.     

On quadratic differentials on multiply connected domains that are perfect squares. II

Problems on extremal decomposition of a multiply connected domain are considered. Two theorems exhibiting the role in such problems of quadratic differentials that are perfect squares are proved. As an implication, it is shown that the extremal metric approach used in the paper naturally leads to the results recently obtained by Dubinin and Eyrikh (Zap. Nauchn. Semin. POMI, 314, 52–75 (2004)) with the use of the capacity theory for generalized condensers. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 40–51.     

Inequalities of Maximum of Partial Sums and Weak Convergence for a Class of Weak Dependent Random Variables

In this paper, we establish a Rosenthal-type inequality of the maximum of partial sums for ρ^- -mixing random fields. As its applications we get the Hájeck -Rènyi inequality and weak convergence of sums of ρ^- -mixing sequence. These results extend related results for NA sequence and p^＊ -mixing random fields,     

Asymptotic expansions forL- andR-statistics under alternatives

In this paper we establish asymptotic expansions (a.e.) under alternatives for the distribution functions of sums of independent identically distributed random variables (i.i.d.r.v.'s.), linear combinations of order statistics, and one-sample rank statistics (L- and R-statistics). The general Lemma from [V. E. Bening,Bull. Moscow State Univ., Ser. 15, 2 36–44 (1994)] is applied to these statistics. Section 1 contains the statement of the theorem, in Sec. 2 the theorems is proved; its proof involves four auxiliary lemmas, also contained in Sec. 2. Finally Sec. 3 contains the proofs of these lemmas. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

Path properties of <Emphasis Type="Italic">l</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-valued Gaussian random fields

General limit theorems are established for l ^p -valued Gaussian random fields indexed by a multidimensional parameter, which contain both almost sure moduli of continuity and limits of large increments for the l ^p -valued Gaussian random fields under Ṅ explicit conditions. This work was supported by NSERC Canada grants at Carleton University and by KOSEF-R01-2005-000-10696-0     

Row and Column Removal Theorems for Homomorphisms of Specht Modules and Weyl Modules

We prove a q-analogue of the row and column removal theorems for homomorphisms between Specht modules proved by Fayers and the first author [16]. These results can be considered as complements to James and Donkin’s row and column removal theorems for decomposition numbers of the symmetric and general linear groups. In this paper we consider homomorphisms between the Specht modules of the Hecke algebras of type A and between the Weyl modules of the q-Schur algebra.This research was supported by ARC grant DP0343023. The first author was also supported by a Sesqui Research Fellowship at the University of Sydney.