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On Q-independence, limit theorems and q-Gaussian distribution

Marcin Marciniak (1998)

Studia Mathematica

We formulate the notion of Q-independence which generalizes the classical independence of random variables and free independence introduced by Voiculescu. Here Q stands for a family of polynomials indexed by tiny partitions of finite sets. The analogs of the central limit theorem and Poisson limit theorem are proved. Moreover, it is shown that in some special cases this kind of independence leads to the q-probability theory of Bożejko and Speicher.

On random split of the segment

Milena Bieniek, Dominik Szynal (2005)

Applicationes Mathematicae

We consider a partition of the interval [0,1] by two partition procedures. In the first a chosen piece of [0,1] is split into halves, in the second it is split by uniformly distributed points. Initially, the interval [0,1] is divided either into halves or by a uniformly distributed random variable. Next a piece to be split is chosen either with probability equal to its length or each piece is chosen with equal probability, and then the chosen piece is split by one of the above procedures. These...

On Riesz product measures ; mutual absolute continuity and singularity

Shelby J. Kilmer, Sadahiro Saeki (1988)

Annales de l'institut Fourier

We give some criteria for mutual absolute continuity and for singularity of Riesz product measures on locally compact abelian groups. The first section gives the definition of such a measure which is more general than the usual definition. The second section provides three sufficient conditions for one Riesz product measure to be absolutely continuous with respect to another. One of our results contains a theorem of Brown-Moran-Ritter as a special case. The final section deals with random Riesz...

On small deviations of Gaussian processes using majorizing measures

Michel J. G. Weber (2012)

Colloquium Mathematicae

We give two examples of periodic Gaussian processes, having entropy numbers of exactly the same order but radically different small deviations. Our construction is based on Knopp's classical result yielding existence of continuous nowhere differentiable functions, and more precisely on Loud's functions. We also obtain a general lower bound for small deviations using the majorizing measure method. We show by examples that our bound is sharp. We also apply it to Gaussian independent sequences and...

On some limit distributions for geometric random sums

Marek T. Malinowski (2008)

Discussiones Mathematicae Probability and Statistics

We define and give the various characterizations of a new subclass of geometrically infinitely divisible random variables. This subclass, called geometrically semistable, is given as the set of all these random variables which are the limits in distribution of geometric, weighted and shifted random sums. Introduced class is the extension of, considered until now, classes of geometrically stable [5] and geometrically strictly semistable random variables [10]. All the results can be straightforward...

On Special Case of Multiple Hypotheses Optimal Testing for Three Differently Distributed Random Variables

Navaei, Leader (2011)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 62P30.In this paper by using theory of large deviation techniques (LDT), the problem of hypotheses testing for three random variables having different distributions from three possible distributions is solved. Hypotheses identification for two objects having different distributions from two given probability distributions was examined by Ahlswewde and Haroutunian. We noticed Sanov's theorem and its applications in hypotheses testing.

On strong laws for generalized L-statistics with dependent data

David Gilat, Roelof Helmers (1997)

Commentationes Mathematicae Universitatis Carolinae

It is pointed out that a strong law of large numbers for L-statistics established by van Zwet (1980) for i.i.d. sequences, remains valid for stationary ergodic data. When the underlying process is weakly Bernoulli, the result extends even to generalized L-statistics considered in Helmers et al. (1988).

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