Criteria for the Strong Law of Large Numbers for Some Classes of Second-Order Stationary Processes and Homogeneous Random Fields

Гапошкин, В. Ф.

doi:10.1137/1122034

Cited by 49 publications

(17 citation statements)

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“…Thus (5), (6) can be obtain for any i > 1. Now we describe some properties of rectangles Some connections of the Gaposhkin asymptotic with stationary processes are given in [2]. It is well-known that the convergences in Birkhoff ergodic theorem and in Kolmogorov S.L.L.N.…”

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confidence: 99%

On the Almost Sure Convergence of All Conditionings of Positive Random Variables

Paszkiewicz¹

2001

Demonstratio Mathematica

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Abstract. We prove that in a non-atomic probability space, for a sequence of positive r.v. (Xn), E(An|2l) -• 0 a.s. for any cr-field 21 of events iff Xn -> 0 a.s. and Esupn>1 |Xn| < 00. We point out these classical theorems on orthogonal series and ergodic means in which the almost sure convergence of all conditionings of a discussed sequence can be obtained. Introduction and main resultsLet 21 be any cr-field of events in any probability space. The operation of conditional expectation E(-|2l) is a positive contraction in the space L\ of integrable random variables. Thus the following theorem can be immediately obtained. Developping such an idea, we obtain the following general THEOREM 1.3. For any non-atomic probability space P) and any sequence (Xi) of positive random variables, the following conditions are equivalent: for any a-fieldThe assumption that (fi, T, P) is non-atomic is essential. Namely,for LO taken from any atom A of 21 with probability P(A) = p, and we have Some consequences of Theorem 1.3 and of (elementary) Theorem 1.1 are collected in Section 3. All these corollaries are obtained by analysing classical proves of specific theorems on almost sure convergence. The proof of Theorem 1.3We shall use the following, rather obvious

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confidence: 99%

On the Almost Sure Convergence of All Conditionings of Positive Random Variables

Paszkiewicz¹

2001

Demonstratio Mathematica

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“…We can omit rather long but standard elementary calculations. We can follow, to some extend, the way indicated by Gaposhkin [2]. From (9) and (22) Theorem follows.…”

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confidence: 87%

Few Remarks on Individual Ergodic Theorem and Summability Methods

Jajte¹

2001

Demonstratio Mathematica

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“…The proof will be reduced to a version of Gaposhkin's criterion for almost sure convergence [8], [9], [2], which in our context says that (3) is equivalent to…”

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confidence: 99%

Pointwise limit theorem for a class of unbounded operators in L^r-spaces

Jajte¹

2007

Studia Math.

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Abstract. We distinguish a class of unbounded operators in L r , r ≥ 1, related to the self-adjoint operators in L 2 . For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin's criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in L r -spaces are applied.1. Introduction. The paper is an attempt of extension of the pointwise ergodic theory in L r (µ)-spaces [10] to the case of some unbounded operators. Some questions are maybe touched here for the first time and perhaps new techniques will have to be developed. We start by trying to follow some ideas and methods known in the case of L 2 (µ). That is why we confine ourselves to a class of linear maps in L r related to unbounded self-adjoint operators in L 2 . To discuss the asymptotic properties of unbounded operators we pass from Cesàro averages to Borel summability. Let us recall the definition.We say that a sequence x = {ξ k } of numbers (or vectors in a Banach soace) is summable to ξ by the Borel method if

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Criteria for the Strong Law of Large Numbers for Some Classes of Second-Order Stationary Processes and Homogeneous Random Fields

Cited by 49 publications

References 5 publications

On the Almost Sure Convergence of All Conditionings of Positive Random Variables

On the Almost Sure Convergence of All Conditionings of Positive Random Variables

Few Remarks on Individual Ergodic Theorem and Summability Methods

Pointwise limit theorem for a class of unbounded operators in L^r-spaces

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Criteria for the Strong Law of Large Numbers for Some Classes of Second-Order Stationary Processes and Homogeneous Random Fields

Cited by 49 publications

References 5 publications

On the Almost Sure Convergence of All Conditionings of Positive Random Variables

On the Almost Sure Convergence of All Conditionings of Positive Random Variables

Few Remarks on Individual Ergodic Theorem and Summability Methods

Pointwise limit theorem for a class of unbounded operators in Lr-spaces

Contact Info

Product

Resources

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Pointwise limit theorem for a class of unbounded operators in L^r-spaces