Precise asymptotics in the Baum-Katz and davis law of large numbers for positively associated sequences

§ 1　 Introduction and resultsL et { X,Xi;i≥ 1} be a sequence of i.i.d.random variables,and set Sn= ni=1 Xi,n≥1.Hsu and Robbins[1 ] introduced the conceptof complete convergence.They together withErdos[2 ] proved n≥ 1 P(|Sn|≥εn) <∞ ,ε>0 (1)if and only if EX=0 and EX2 <∞ .L ater,Spitzer[3] proved n≥ 11n P(|Sn|≥εn) <∞ ,ε>0if and only if EX =0 and E|X|<∞ .More generally,it was shown by Baum and Katz[4 ]that,for 0 0 (…     

A BMAP|G|1-analysis based on convolution calculus

Stochastic models of high-speed communication systems typically require the analysis of special Markov additive processes, known thus far as “batch Markovian arrival processes” (BMAPs). In most cases, the solution of nonlinear matrix equations of the form or is necessary, leading, in general, to severe numerical problems. Several efficient algorithms for that purpose have been reported in the literature. In this contribution, we present a new approach (the BMAP|G| 1-analysis), which is based on a convolution calculus for matrix sequences. The main results are the unified representation of Poisson, compound Poisson, and batch Markovian arrival processes by so-called convolutional exponential distributions, and the computation of the fundamental-period matrix G via so-called semi-convolutions. The resulting algorithms, in general, cannot outperform known algorithms as, for instance, those exploiting Horner schemes as proposed by D. M. Lucantoni and other authors, or the highly efficient cyclic reduction algorithm of D. Bini and B. Meini; nevertheless, they allow the direct computation of step matrices such as the “arrival matrices” A_n with and the component matrices G_n summing to G. Further, there is some hope that the proposed framework may lead to a unification of the theory in the sense that queues with arrival and/or service processes that are characterized by convolutional exponential distributions may possess formally similar properties, as known from the scalar case. Proceedings of the Seminar on Stability Problems for Stochastic Models. Hajdúszoboszló. Hungary, 1997. Part II.     

Carleson measures and the heat equation

Let { }, where { } is the open unit disk on the complex plane { }. In G, we consider analytic solutions u(t, z) ({ }, { }) of the heat equation 2u_t=u_zz with initial data f(z)=u(0, z) belonging to the Fock space F, i.e., to the space of entire functions square summable with the weight e−|z|2.Conditions on a nonnegative measure μ on G are described under which for all f ∈ F we have { } Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 146–155. Translated by S. V. Kislyakov.     

The central limit theorem for geodesic flows onn-dimensional manifolds of negative curvature

In this paper we prove a central limit theorem for special flows built over shifts which satisfy a uniform mixing of type , 0<γ<1, α>0. The function defining the special flow is assumed to be continuous with modulus of continuity of type , 0<ρ<1, β>0, andd is the natural metric on sequence space. Geodesic flows on compact manifolds of negative curvature are isomorphic to special flows of this kind.     

On the order of partial sums of general orthogonal series

It is shown that for convergence of every orthonormal system _n(x) given on [0, l],it is necessary and sufficient that, under the condition on tlie increasing function W(x) and for there hold almost everywhere on [0, 1].Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 451–462, October, 1969.     

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.     

Dirichlet series whose partial sums of coefficients have regular variation

We study pairs of Dirichlet series and in whicha(n) counts the number of objects of “size”n of some class of objects which is closed under formation of direct products and extraction of irreducible factors, andp(n) counts the number of objects of “size”n which are irreducible in this class. We prove Dirichlet series analogues of certain results about power series and use these results to prove some conjectures of Burris concerning first-order 0–1 laws.     

On a certain basis inc 0

A basis is constructed inc ₀ such that there exists no bounded linear projection ofc ₀ onto the subspace spanned by a certain subsequence of . This is part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the suppervision of Professor A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank Dr. Lindenstrauss for his helpful advice.     

Identities Involving Coefficients of Automorphic L-Functions

Let f(z) be a holomorphic Hecke eigenform of weight k with respect to SL(2, ℤ) and let

Invariants of classCk for finite coxeter groups and their representations in terms of anisotropic spaces

This article is devoted to the study of representations of C^k { } functions f invariant with respect to finite Coxeter groups W in the form f=F op, where p is a base in the algebra of W-invariant polynomials. We examine the lowering of smoothness of F as compared with f and conclude that this lowering has an anisotropic nature and that, more precisely, at each point Po it is described by a vector { }. We examine the cases W=A_n, B_n, D_n, { }; in each case the greatest component { } of { } is equal to the Coxeter number of the stabilizer W_yo of the point yo, where po=p(yo). Bibliography: 22 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 46–70. Translated by S. Yu. Pilyugin.     

Inequalities of Paley type for noncommutative martingales

Summary The purpose of this paper is to study the validity of the Paley inequality on square function, for noncommutative martingales. Let be a regular gage space, and a sequence of von-Neumann algebras such that we prove that for every , where ɛ_n(F) is the conditional expectation of F with respect to the subalgebra : We also consider the case of a martingale arising in the context of harmonic analysis on noncommutative discrete groups, in analogy to the theorem of R.E.A.C. Paley on Fourier-Walsh series. Entrata in Redazione il 26 gennaio 1977. Partially sponsored by C.N.R.     

An interesting result for positive series

For arbitrary given positive term series and , the present paper provides an interesting necessary and sufficient condition for . In Memory of Professor M. T. Cheng The Research Project of the Mathematical Institute, Ningbo University. Supported in part by Zhejiang Provincial Natural Science Foundation, and by State Key Laboratory of Southwest Institute of Petroleum.     

On the LIL for Self-Normalized Sums of IID Random Variables

Let be i.i.d. random variables and let, for each and . It is shown that a.s. whenever the sequence of self-normalized sums S _n /V _n is stochastically bounded, and that this limsup is a.s. positive if, in addition, X is in the Feller class. It is also shown that, for X in the Feller class, the sequence of self-normalized sums is stochastically bounded if and only if      

Коэффициенты рядов Ф урье и класс Липшица

We generalize and sharpen certain results concerning Fourier series from the Lipschitz class. In particular, for sinnx we prove the following: Let ¦b_n¦n^–2L(n) where L(x) is a continuous and slowly oscillating function. Then

A local duality theorem for categories of modules

Let Λ be an associative ring with identity and let be the category of left unitary Λ-modules. A subcateqory of the category is said to be small if the pairwise nonisomorphic objects of form a set. The main result of this paper consists of the fact that for every small full subcategory , there exists a ring Γ such that is dual to a small full subcategory of the category . Some applications of this result are indicated. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 66–73.     

On the numerical construction of an attractor for the Navier-Stokes system

In this paper the problem of the numerical approximation of the minimal global B-attractor for a semiflow generated by the Navier-Stokes equations in a two-dimensional bounded domain Ω is considered. The method suggested here is based on the formula , where G^N is a sequence of compact subsets of L₂(Ω), . The procedure of constructing G^N is finite and includes the numerical solution of the Navier-Stokes equations by means of the Galerkin method, together with an explicit finite-difference discretization in time. Bibliography: 5 titles. To dear teacher Olga A. Ladyzhenskaya on the occasion of the jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 91–97. Translated by V. I. Ochkur.     

Strong law of large numbers for functionals of products of random matrices

For the sequence A₁, A₂, ..., A_n, ... of m×m independent random matrices such that for each k there exists a joint density function P_k(X) of the elements _ij ^k, we prove the following theorem: if and for some positive constants ₁ and ₂, then with probability 1,

Lacunary series and pseudocontinuations

The main goal of this paper is to prove the following statement. Let be a function holomorphic and of bounded characteristic in the unit disk , where E is a Λ(1)-subset of ℤ₊. Assume that f has a pseudocontinuation of bounded characteristic to the annulus {zɛℂ:1<|z|<R}. Then f admits analytic continuation to the disk R . In particular, f is a polynomial if R=+∞. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 16–32. Translated by A. B. Aleksandrov.