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Few remarks on individual ergodic theorem and summability methods

100%

Jajte R.

Demonstratio Mathematica

2001

tom Vol. 34, nr 2

315-320

Individual ergotic theorem for non-contractive normal operators

100%

Jajte R.

Probability and Mathematical Statistics

2002

tom Vol. 22, Fasc. 1

85--99

A condition implying the strong law of large numbers for trajectories of a normal non-contractive operator is given. The condition has been described in terms of a spectral measure, in the spirit of the well-known theorem of V. F. Gaposhkin. To embrace the non-contractive operators we pass from the classical arithmetic (Cesáro) means to the Borel methods of summability.

Exponential formula related to the Gauss semigroup

100%

Jajte R.

Probability and Mathematical Statistics

2006

tom Vol. 26, Fasc. 1

113--120

For the second derivative, an analogue of the classical Taylor’s formula is considered on a suitable function space. The sum of the ‘Taylor series” represents the Gauss semigroup. This may be useful in describing the trajectories of some functions under the action of the Gauss semigroup.

Pseudo-martingales

63%

Jajte R. , Paszkiewicz A.

Probability and Mathematical Statistics

1999

tom Vol. 19, Fasc. 1

181--201

For a probability space (Ω, ℱ, P)and a filtration (Un) in Ω,we consider the sequences (Xn) of random variables satisfying the condition E (Xn+1 − Xn | Un) = 0, n = 1, 2, ... In general, the proces (Xn) is not required to be (Un) adapted and it is called a pseudo-martingale. We indicate simple and natural conditions implying a good asymptotic behaviour of pseudo-martingales. For example: let (Xn, Un) be a uniformly integrable pseudo-martingale with Un ↗ ℱ. Then Xn → X weakly in L1 where [formula]. Some approximation results for σ-fields are obtained with implications to pseudo-martingales. A number of examples is collected.

Conditioning and weak convergence

63%

Jajte R. , Paszkiewicz A.

Probability and Mathematical Statistics

1999

tom Vol. 19, Fasc. 2

453--461

Several connections between the weak convergence of random variables, convergence of their distributions and conditioning have been described.

Weak convergence of random vectors and distributions in Banach spaces

63%

Jajte R. , Paszkiewicz A.

Probability and Mathematical Statistics

2002

tom Vol. 22, Fasc. 1

1--11

Let (ξn) be a sequence of random vectors with values in a Banach space X with distributions pξn weakly converging to a given distribution p. We characterize a general form of a distribution of a weak limit of ξ n in Banach space L1(X) of Bochner integrable vectors. We show that the weak convergence of random vectors (ξn) in L1(X) implies that ‖ξn(ω)−ξ(ω)‖ →0 stochastically. Moreover, the conditions ‖ξn(ω)−ξ(ω)‖ →0 stochastically and〈ξn(ω)−ξ(ω), x*〉→0 stochastically for any x*∈X*are

Random stain

51%

Hensz-Chądzyńska E. , Jajte R. , Paszkiewicz A.

Probability and Mathematical Statistics

1998

tom Vol. 18, Fasc. 1

199--218

A stochastic model of a stain of pollution is proposed. The asymptotic shape of the edges of a stain is examined. The description is kept at the elementary level of probability theory.