Modern Theory of Summation of Random Variables

Золотарев, В. М.

doi:10.1515/9783110936537

Cited by 205 publications

(144 citation statements)

References 0 publications

Supporting

Mentioning

142

Contrasting

Order By: Relevance

“…We follow the monograph [4] when discussing the history of the topic. The metric approach to the problems on the accuracy of approximations of distributions appeared in the theory of probability in the mid 1930s.…”

Section: Discussion Of Results and Some Historical Remarksmentioning

confidence: 99%

See 1 more Smart Citation

The convergence of $U$-statistics and von Mises functionals in the mean metric

Sharakhmetov

2004

Theor. Probability and Math. Statist.

View full text Add to dashboard Cite

Abstract. We prove global limit theorems for U -statistics and von Mises functionals. The rate of convergence in these theorems is also obtained.Limit theorems for U -statistics are usually considered for the uniform metric (see [1]-[3]). This paper is devoted to limit theorems for the mean metric for which we study the convergence of distributions of U -statistics and von Mises functionals.We prove global limit theorems and obtain the rate of convergence in these theorems for both degenerate and nondegenerate kernels.The idea of the proofs of the results is based on the method of metric distances (see [4]). There are other approaches used in the theory of U -statistics, such as the martingale method, the method of characteristic functions, the method of orthogonal decompositions, and some others. It is worthwhile to mention that V. M. Zolotarev ([4, p. 404]) was the first to point out that the method of metric distances works in the case of nonlinear models, too.The results of the paper are announced in [5].

show abstract

Section: Discussion Of Results and Some Historical Remarksmentioning

confidence: 99%

“…The idea of the proofs of the results is based on the method of metric distances (see [4]). There are other approaches used in the theory of U -statistics, such as the martingale method, the method of characteristic functions, the method of orthogonal decompositions, and some others.…”

mentioning

confidence: 99%

The convergence of $U$-statistics and von Mises functionals in the mean metric

Sharakhmetov

2004

Theor. Probability and Math. Statist.

View full text Add to dashboard Cite

show abstract

“…The latter definition is a version of the minimal (or coupling) distance between the probability distributions; see, for example, [26] or [27], Section 11.8.…”

Section: Definition 22mentioning

confidence: 99%

Extended Poisson equation for weakly ergodic Markov processes

Veretennikov

Kulik

2013

Theor. Probability and Math. Statist.

View full text Add to dashboard Cite

Abstract. Solvability conditions for a Poisson equation with an extended generator of a general Markov process are obtained. The predictable part in the Doob-Meyer decomposition is described for a process of the form g(X(t), Y (t)), where Y is a solution of a stochastic equation with the coefficients depending on X and where the function g = g(x, y) is defined as a family of solutions of the Poisson equation.

show abstract

“…This function is an ideal probability metric of order s = 1 (see [15], Example 1.4.2.) and note that (E, µ) is a metric space.…”

Section: Examplesmentioning

confidence: 99%

On some definition of expectation of random element in metric space

Bator

Zięba

2009

Annales UMCS, Mathematica

View full text Add to dashboard Cite

Abstract. We are dealing with definition of expectation of random elements taking values in metric space given by I. Molchanov and P. Teran in 2006. The approach presented by the authors is quite general and has some interesting properties. We present two kinds of new results:• conditions under which the metric space is isometric with some real Banach space; • conditions which ensure "random identification" property for random elements and almost sure convergence of asymptotic martingales.

show abstract

Modern Theory of Summation of Random Variables

Cited by 205 publications

References 0 publications

The convergence of $U$-statistics and von Mises functionals in the mean metric

The convergence of $U$-statistics and von Mises functionals in the mean metric

Extended Poisson equation for weakly ergodic Markov processes

On some definition of expectation of random element in metric space

Contact Info

Product

Resources

About