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1

Chain dependent continuous time random walk

Szczotka W., Żebrowski P.

Probability and Mathematical Statistics

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2011

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Vol. 31, Fasc. 2

239--261

EN

An asymptotic behavior of a continuous time random walk is investigated in the case when the sequence of pairs of jump vectors and times between jumps is chain dependent.

2

On the remarkable distributions of maxima of some fragments of the standard reflecting random walk and Brownian motion

Fujita T., Yor M.

Probability and Mathematical Statistics

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2007

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Vol. 27, Fasc. 1

89--104

EN

In this paper, we consider some distributions of maxima of excursions and related variables for standard random walk and Brownian motion. We discuss the infinite divisibility properties of these distributions and calculate their Lévy measures. Lastly we discuss Chung's remark related with Riemann's zeta functional equation.

3

Lévy processes and stochastic integrals in Banach spaces

Applebaum D.

Probability and Mathematical Statistics

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2007

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Vol. 27, Fasc. 1

75--88

EN

We review infinite divisibility and Lévy processes in Banach spaces and discuss the relationship with notions of type and cotype. The Lévy-Itô decomposition is described. Strong, weak and Pettis-style notions of stochastic integral are introduced and applied to construct generalised Ornstein-Uhlenbeck processes

4

On harmonic measure for Lévy processes

Sztonyk P.

Probability and Mathematical Statistics

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2000

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Vol. 20, Fasc. 2

383--390

EN

Let {Xt} be a Lévy process in Rd, d ≥ 2, with infinite Lévy measure. If {Xt} has no Gaussian component, then the process does not hit the boundary of Lipschitz domain S ⊂ Rd at the first exit time of S under mild conditions on {Xt}. The conditions are met, e.g., if {Xt} is rotation invariant.

5

Some multivariate infinitely divisible distributions and their projections

Maejima M., Suzuki K., Tamura Y.

Probability and Mathematical Statistics

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1999

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Vol. 19, Fasc. 2

421--428

EN

Recently K. Sato constructed an infinitely divisible probability distribution μ on Rd such that μ is not selfdecomposable but every projection of μ to a lower dimensional space is selfdecomposablc. Let Lm (Rd), 1 ≤ m < ∞, be the Urbanik-Sato type nested subclasses of the class L0 (Rd) of all selfdecomposable distributions on Rd. In this paper, for each 1 ≤ m < ∞, a probability distribution μ with the following properties is constructed: μ belongs to Lm-1 (Rd) ∩ (Lm (Rd))c, but every projection of μ to a lower k-dimensional space belongs to Lm (Rk). It is also shown that Sato's example is not only "non-selfdecomposable" but also "non-semi-selfdecomposable."

6

Point regularity of p-stable density in Rd and Fisher information

Lewandowski M.

Probability and Mathematical Statistics

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1999

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Vol. 19, Fasc. 2

375--388

EN

In the paper we prove that the n-th directional derivative of a p-stable density f (x) in the direction a can be estimated by [formula], where 0 < u < 1, and C depends also on geometrical properties of the Lévy measure. This inequality helps us to calculate the Fisher information of stable measures.

7

A Lévy Khintchine type representation of convolution semigroups : on commutative hypergroups

Rentzsch C.

Probability and Mathematical Statistics

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1998

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Vol. 18, Fasc. 1

185--198

EN

We first study Lévy measures, Poisson and Gaussian convolution semigroups on commutative hypergroups. Then we present a Lévy-Khintchine type representation of a convolution semigroup (μt)t>0with symmetric Lévy measure λ of the form μt = γt,*e(tγ), t≥0, for some Poisson semigroup (e(tγ)) t>0 and some Gaussian semigroup (γt) t>0.