Lacunary tangential approximation

Our aim is to give lacunary versions with upper density equal to the value one of Arakelian's approximation theorem for special geometries of the domain and the closed set.By use of the theorems, we are able to construct functions which have so-called maximal cluster sets along arbitrary curves and which have a lacunary structure in addition. Upper density equal to the value one will turn out to be best possible for this result.     

Almost sure central limit theorems for random fields

A general almost sure limit theorem is presented for random fields. It is applied to obtain almost sure versions of some (functional) central limit theorems. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some applications of the Knaster-Kuratowski and Mazurkiewicz principle in hyperconvex metric spaces

In this paper, as applications of the Knaster-Kuratowski and Mazurkiewicz principle in the hyperconvex version, we obtain the Ky Fan type matching theorems for closed and open covers. As applications, some intersection theorems which are hyperconvex versions of corresponding results due to Alexandroff and Pasynkoff, Fan, Klee, Horváth, and Lassonde are established. Then, the Ky Fan type best approximation theorem and Schauder-Tychonoff fixed-point theorem (i.e., Fan-Glicksberg fixed-point theorem) for set-valued mappings in noncompact hyperconvex spaces are also given. Finally, we obtain a general form of the Browder-Fan fixed-point theorem for set-valued mappings in noncompact hyperconvex spaces. These results include corresponding results in the literature as special cases.     

Random Coincidence Point Theorem in Fréchet Spaces with Applications

Abstract

We proved a random coincidence point theorem for a pair of commuting random operators in the setup of Fréchet spaces. As applications, we obtained random fixed point and best approximation results for *-nonexpansive multivalued maps. Our results are generalizations or stochastic versions of the corresponding results of Shahzad and Latif [Shahzad, N.; Latif, A. A random coincidence point theorem. J. Math. Anal. Appl. 2000, 245, 633–638], Khan and Hussain [Khan, A.R.; Hussain, N. Best approximation and fixed point results. Indian J. Pure Appl. Math. 2000, 31 (8), 983–987], Tan and Yaun [Tan, K.K.; Yaun, X.Z. Random fixed point theorems and approximation. Stoch. Anal. Appl. 1997, 15 (1), 103–123] and Xu [Xu, H.K. On weakly nonexpansive and *-nonexpansive multivalued mappings. Math. Japon. 1991, 36 (3), 441–445].     

A minimax theorem for functions with possibly nonconnected intersections of sublevel sets

We apply the selection theorem for multivalued mappings with paraconvex values (rather than various versions of KKM-principle) to prove several minimax theorems. In contrast with well-known minimax theorems for coordinatewise semicontinuous functions, in our theorems finite intersections of sublevel or uplevel sets can be nonempty and nonconnected.     

Random fixed point theorems for composites of acyclic multifunctions

In this paper we obtain random versions of Kakutani–Fan type fixed point theorems for a class V _c ⁺ of multifunctions which contains Kakutani factorizable maps and composites of acyclic maps. As applications, we derive some random approximation theorems.     

Stability in first approximation of stochastic systems with after effect : PMM vol. 40, n 6, 1976, pp. 1116–1121

The theorem on existence of the Liapunov functionals and the theorem on stability in first approximation for a stochastic differential equation with aftereffect are proved.The suggestion of the replacement of Liapunov functions by functionals [1] in the investigation of the stability of ordinary differential equations with lag, has been widely utilized in dealing with determinate systems, as well as in the case of linear and nonlinear stochastic systems (see e. g. [2 – 11]). Results concerning the stability in the first approximation were obtained for stochastic systems in [12 – 18] and others. Use of Liapunov functionals for the differential equations with aftereffect was first encountered in [1, 19, 20] where the inversion theorems were proved and conditions for the stability in first approximation were obtained.Below a stochastic differential equation with aftereffect is investigated where the random perturbations represent an arbitrary process with independent increments.     

任意随机序列关于非齐次马氏链的随机和的一类随机偏差定理

In this paper, the notion of limit random logarithmic likelihood ratio of stochastic sequence, as a measure of dissimilarity between the joint distribution on measure P and the Markov distribution on measure Q, is introduced. A class of random approximation theorems for arbitrary stochastic dominated sequence are obtained by using the tools of generating functions and the tailed-probability generating functions.     

Direct and converse theorems of the theory of approximation in the metric of Lp for 0 < p < 1

A direct theorem of the Jackson type and several converse theorems are established for the approximation of periodic functions of period 2Π by trigonometrical polynomials in the metric of L_p, 0 < p < 1.     

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.     

Integral analogues of almost sure limit theorems

Summary An integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable processes, moreover for processes being in the domain of attraction of a stable law or being in the domain of geometric partial attraction of a semistable or a max-semistable law.     

Convergence analysis of the general Gauss-Newton algorithm

Summary The convergence of the Gauss-Newton algorithm for solving discrete nonlinear approximation problems is analyzed for general norms and families of functions. Aquantitative global convergence theorem and several theorems on the rate of local convergence are derived. A general stepsize control procedure and two regularization principles are incorporated. Examples indicate the limits of the convergence theorems.     

Runge approximation theorems in complex Clifford analysis together with some of their applications

A number of Runge approximation theorems are proved for complex Clifford algebra valued holomorphic functions which either satisfy the holomorphic, homogeneous Dirac equation, or complex Laplacian. The results are applied to establish analogues of the homological version of the Mittag-Leffler theorem.     

Approximation theorems by positive linear operators in weighted spaces

In this study, we obtain some Korovkin type approximation theorems by positive linear operators on the weighted space of all real valued functions defined on the real two-dimensional Euclidean space \mathbbR²{\mathbb{R}^2}. This paper is mainly consisted of two parts: a Korovkin type approximation theorem via the concept of A-statistical convergence and a Korovkin type approximation theorem via A{\mathcal {A}}-summability.     

关于相依离散随机序列的若干强偏差定理

首次引入随机序列滑动似然比与滑动相对熵概念作为刻画任意相依随机序列的独立逼近的随机性度量,利用B-C引理与分析方法相结合,研究任意离散随机序列部分和滑动平均的强偏差定理.作为推论,得到了关于广义经验分布函数的一个极限定理.最后,给出了若干例子.     

对数似然比与离散随机变量序列强极限定理的一种分析方法

本文引进对数似然比作为任意离散随机变量序列相依性的一种度量，并通过限制似然比给出样本空间的某种子集，在这种子集上得到了离散随机变量序列的一类强极限定理，它包含若干经典强大数定律为其特例．在证明中本文提出了证明强极限定理的一种分析方法，其要点是将关于单调函数可微性的定理应用于几乎处处收敛的研究．     

Joint discrete universality for periodic zeta-functions

Abstract

In the paper, joint universality theorems for periodic zeta functions with multiplicative coefficients and periodic Hurwitz zeta-functions are proved. The main theorem of [11] is extended, and two new joint universality theorems on the approximation of a collection of analytic functions by discrete shifts of the above zeta-functions are obtained. For this, certain linear independence hypotheses are applied.     

Addition theorems for Legendre functions as corollaries of an addition theorem for Gegenbauer polynomials

Proofs are given for addition theorems for Legendre functions with arbitrary upper and lower indices, based solely on an addition theorem for Gegenbauer polynomials. New versions of these and other similar theorems are given, both in the form of sums and of integrals.Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 321–330, September, 1976.     

Stein's method and random character ratios

Stein's method is used to prove limit theorems for random character ratios. Tools are developed for four types of structures: finite groups, Gelfand pairs, twisted Gelfand pairs, and association schemes. As one example an error term is obtained for a central limit theorem of Kerov on the spectrum of the Cayley graph of the symmetric group generated by -cycles. Other main examples include an error term for a central limit theorem of Ivanov on character ratios of random projective representations of the symmetric group, and a new central limit theorem for the spectrum of certain random walks on perfect matchings. The results are obtained with very little information: a character formula for a single representation close to the trivial representation and estimates on two step transition probabilities of a random walk. The limit theorems stated in this paper are for normal approximation, but many of the tools developed are applicable for arbitrary distributional approximation.