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Maejima M., Tudor C. A.

Probability and Mathematical Statistics

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2012

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Vol. 32, Fasc. 1

167--186

EN

We study selfsimilar processes with stationary increments in the second Wiener chaos. We show that, in contrast with the first Wiener chaos which contains only one such process (the fractional Brownian motion), there is an infinity of selfsimilar processes with stationary increments living in the Wiener chaos of order two. We prove some limit theorems which provide a mechanism to construct such processes.

2

Several forms of stochastic integral representations of gamma random variables and related topics

Aoyama T., Maejima M., Ueda Y.

Probability and Mathematical Statistics

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2011

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Vol. 31, Fasc. 1

99--118

EN

Gamma distributions can be characterized as the laws of stochastic integrals with respect to many different Lévy processes with different nonrandom integrands. A Lévy process corresponds to an infinitely divisible distribution. Therefore, many infinitely divisible distributions can yield a gamma distribution through stochastic integral mappings with different integrands. In this paper, we pick up several integrands which have appeared in characterizing well-studied classes of infinitely divisible distributions, and find inverse images of a gamma distribution through each stochastic integral mapping. As a by-product of our approach to stochastic integral representations of gamma random variables, we find a remarkable new general characterization of classes of infinitely divisible distributions, which were already considered by James et al. (2008) and Aoyama et al. (2010) in some special cases.

3

The class of type G distributions of Rd and related subclasses of infinitely divisible distribution

Maejima M., Rosiński J.

Demonstratio Mathematica

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2001

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Vol. 34, nr 2

251-266

EN

Classes of infinitely divisible distributions obtained by iteration of Gaus-sian randomization of Levy measures are introduced and studied. Their relation to Urbanik-Sato nested classes of selfdecomposable distributions is also established.

4

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."