On Equivalence of Infinite Product Measures

Kakutani, Shizuo

doi:10.2307/1969123

Cited by 409 publications

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“…This result is a dichotomy property for this kind of measure, and its proof relies on a convergence theorem for products of random variables, which is essentially a generalization of Kakutani's famous dichotomy criterion for infinite product measures [4]. As G. Brown and W. Moran [2] pointed out, the measure \i is, in this case, a Riesz product-type measure of the form…”

Section: Introductionmentioning

confidence: 93%

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L^p-convergence of a certain class of product martingales

Koumandos¹

1994

Proceedings of the Edinburgh Mathematical Society

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Section: Introductionmentioning

confidence: 93%

“…So, this gives reason to investigate analogous dichotomy theorems in this general situation as both Kakutani's criterion [4] and Brown and Moran's convergence theorem (see [2, Theorem 1]) seem to be effective.…”

Section: Introductionmentioning

confidence: 99%

L^p-convergence of a certain class of product martingales

Koumandos¹

1994

Proceedings of the Edinburgh Mathematical Society

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“….) of probability measures on Σ is strongly positive if there is a real number δ > 0 such that, for all n ∈ N and a ∈ Σ, α (n) (a) ≥ δ. Kakutani [10] proved the classical, measure-theoretic version of the following theorem, and van Lambalgen [25,26] and Vovk [27] extended it to algorithmic randomness. Theorem 3.12.…”

Section: Theorem 34 ([15])mentioning

confidence: 99%

Mutual Dimension and Random Sequences

Case

Lutz

2015

Mathematical Foundations of Computer Science 2015

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If S and T are infinite sequences over a finite alphabet, then the lower and upper mutual dimensions mdim(S : T ) and M dim(S : T ) are the upper and lower densities of the algorithmic information that is shared by S and T . In this paper we investigate the relationships between mutual dimension and coupled randomness, which is the algorithmic randomness of two sequences R 1 and R 2 with respect to probability measures that may be dependent on one another. For a restricted but interesting class of coupled probability measures we prove an explicit formula for the mutual dimensions mdim(R 1 : R 2 ) and M dim(R 1 : R 2 ), and we show that the condition M dim(R 1 : R 2 ) = 0 is necessary but not sufficient for R 1 and R 2 to be independently random.We also identify conditions under which Billingsley generalizations of the mutual dimensions mdim(S : T ) and M dim(S : T ) can be meaningfully defined; we show that under these conditions these generalized mutual dimensions have the "correct" relationships with the Billingsley generalizations of dim(S), Dim(S), dim(T ), and Dim(T ) that were developed and applied by Lutz and Mayordomo; and we prove a divergence formula for the values of these generalized mutual dimensions.

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“…In a statistical context these integrals have been introduced by Bhattacharyya (1943). Later Kakutani (1948) stressed the fact that the Hellinger integral is an inner product. The Hellinger integral was again introduced by Matusita (1951) under the name "affinity".…”

Section: Robustness Of Statistical Decision Proceduresmentioning

confidence: 99%

Asymptotically Disjoint Quantum States

Primas

2000

Decoherence: Theoretical, Experimental, and Conceptual Problems

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A clarification of the heuristic concept of decoherence requires a consistent description of the classical behavior of some quantum systems. We adopt algebraic quantum mechanics since it includes not only classical physics, but also permits a judicious concept of a classical mixture and explains the possibility of the emergence of a classical behavior of quantum systems. A nonpure quantum state can be interpreted as a classical mixture if and only if its components are disjoint. Here, two pure quantum states are called disjoint if there exists an element of the center of the algebra of observables such that its expectation values with respect to these states are different. An appropriate automorphic dynamics can transform a factor state into a classical mixture of asymptotically disjoint final states. Such asymptotically disjoint quantum states lead to regular decision problems while exactly disjoint states evoke singular problems which engineers reject as improperly posed.

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On Equivalence of Infinite Product Measures

Cited by 409 publications

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L^p-convergence of a certain class of product martingales

L^p-convergence of a certain class of product martingales

Mutual Dimension and Random Sequences

Asymptotically Disjoint Quantum States

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On Equivalence of Infinite Product Measures

Cited by 409 publications

References 0 publications

Lp-convergence of a certain class of product martingales

Lp-convergence of a certain class of product martingales

Mutual Dimension and Random Sequences

Asymptotically Disjoint Quantum States

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Product

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L^p-convergence of a certain class of product martingales

L^p-convergence of a certain class of product martingales