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1
Content available remote Exponential formula related to the Gauss semigroup
Jajte R.
Probability and Mathematical Statistics
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2006
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Vol. 26, Fasc. 1
113--120
EN
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.
2
Content available remote Weak convergence of random vectors and distributions in Banach spaces
Jajte R., Paszkiewicz A.
Probability and Mathematical Statistics
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2002
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Vol. 22, Fasc. 1
1--11
EN
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
3
Content available remote Individual ergotic theorem for non-contractive normal operators
Jajte R.
Probability and Mathematical Statistics
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2002
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Vol. 22, Fasc. 1
85--99
EN
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.
4
Content available remote Few remarks on individual ergodic theorem and summability methods
Jajte R.
Demonstratio Mathematica
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2001
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Vol. 34, nr 2
315-320
5
Content available remote Pseudo-martingales
Jajte R., Paszkiewicz A.
Probability and Mathematical Statistics
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1999
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Vol. 19, Fasc. 1
181--201
EN
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.
6
Content available remote Conditioning and weak convergence
Jajte R., Paszkiewicz A.
Probability and Mathematical Statistics
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1999
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Vol. 19, Fasc. 2
453--461
EN
Several connections between the weak convergence of random variables, convergence of their distributions and conditioning have been described.
7
Content available remote Random stain
Hensz-Chądzyńska E., Jajte R., Paszkiewicz A.
Probability and Mathematical Statistics
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1998
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Vol. 18, Fasc. 1
199--218
EN
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.
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