Some multivariate infinitely divisible distributions and their projections

• Autorzy: Maejima, M. , Suzuki, K. , Tamura, Y.
• Języki publikacji: EN

Abstrakty

EN

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

Słowa kluczowe

EN

Lévy measure; Lévy process; semi-selfdecomposable distributions; infinitely divisible distributions

• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 1999
• Tom: Vol. 19, Fasc. 2
• Strony: 421--428
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Maejima, M.

• Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

autor

Suzuki, K.

• Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

autor

Tamura, Y.

• Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

Bibliografia

• [1] Z. J. Jurek, The classes Lm (Q) of probability measures on Banach spaces, Bull. Polish Acad. Sci. Math. 31 (1983), pp. 51-62.
• [2] M. Maejima and Y. Naito, Semi-selfdecomposable distributions and a new class of limit theorems, Probab. Theory Related Fields 112 (1998), pp. 13-31.
• [3] G. Samorodnitsky and M. S. Taqqu, Stable Nan-Gaussian Random Processes, Chapman and Hall, New York-London 1994.
• [4] K. Sato, Class L of multivariate distributions and its subclasses, J. Multivariate Anal. 10 (1980) pp. 207-232.
• [5] - Multivariate distributions with selfdecomposable projections, J. Korean Math. Soc. 35 (1998), pp. 783-791.
• [6] - and M. Yamazato, Operator-selfdecomposable distributions us limit distributions of processes of Ornstein-Uhlenbeck type, Stochastic Process. Appl. 17 (1984), pp. 73-100.
• [7] D. N. Shanbhag and M. Sreehari, An extension of Goldie's result and further results in infinite divisibility, Z. Wahrsch. verw. Gebiete 47 (1979), pp. 19-25.
• [8] K. Urbanik, Self-decomposable probability distributions on Rm, Zastos. Mat. 10 (1969), pp. 91-97.
• [9] - Slowly varying sequences of random variables, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), pp. 679-682.
• [10] - Limit laws for sequences of normed sums satisfying some stability conditions, in: Multivariate Analysis-III, P. R. Krishnaiah (Ed.), Academic Press, New York 1973, pp. 225-237.