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Strong law of large numbers for additive extremum estimators

100%
Mexia J., Real P.
Discussiones Mathematicae Probability and Statistics
|
2001
|
tom 21
|
nr 2
81-88
EN
Extremum estimators are obtained by maximizing or minimizing a function of the sample and of the parameters relatively to the parameters. When the function to maximize or minimize is the sum of subfunctions each depending on one observation, the extremum estimators are additive. Maximum likelihood estimators are extremum additive whenever the observations are independent. Another instance of additive extremum estimators are the least squares estimators for multiple regressions when the usual assumptions hold. A strong law of large numbers is derived for additive extremum estimators. This law requires only the existence of first order moments and may be of interest in connection with maximum likelihood estimators, since the usual assumption that the observations are identically distributed is discarded.
2
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On strong uniform distribution, II. The infinite-dimensional case

88%
Lacroix Y.
Acta Arithmetica
|
1998
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tom 84
|
nr 3
279-290
EN
We construct infinite-dimensional chains that are L¹ good for almost sure convergence, which settles a question raised in this journal [N]. We give some conditions for a coprime generated chain to be bad for L² or $L^∞$, using the entropy method. It follows that such a chain with positive lower density is bad for $L^∞$. There also exist such bad chains with zero density.
3
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Almost sure approximation of unbounded operators in $L_2 (X,A,μ)$

75%
Jajte R., Paszkiewicz A.
Studia Mathematica
|
1998
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tom 128
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nr 2
103-120
EN
The possibilities of almost sure approximation of unbounded operators in $L_2(X,A,μ)$ by multiples of projections or unitary operators are examined.
4
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On the bundle convergence of double orthogonal series in noncommutative $L_2$-spaces

63%
Móricz F., Le Gac B.
Studia Mathematica
|
2000
|
tom 140
|
nr 2
177-190
EN
The notion of bundle convergence in von Neumann algebras and their $L_2$-spaces for single (ordinary) sequences was introduced by Hensz, Jajte, and Paszkiewicz in 1996. Bundle convergence is stronger than almost sure convergence in von Neumann algebras. Our main result is the extension of the two-parameter Rademacher-Men'shov theorem from the classical commutative case to the noncommutative case. To our best knowledge, this is the first attempt to adopt the notion of bundle convergence to multiple series. Our method of proof is different from the classical one, because of the lack of the triangle inequality in a noncommutative von Neumann algebra. In this context, bundle convergence resembles the regular convergence introduced by Hardy in the classical case. The noncommutative counterpart of convergence in Pringsheim's sense remains to be found.
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