# zbMATH — the first resource for mathematics

Operator geometric stable laws. (English) Zbl 1069.60017
Operator geometric stable laws (OGS) on a vector space $$\mathbb{R}^d$$ are limits of randomized affine normalized sums $$A_p\sum_1^{N_p}(X_i + b_b)$$, $$p\to 0$$, where $$X_i$$ are iid and $$N_p$$ is a geometrically distributed random variable (with mean $$1/p$$) which is independent from $$X_i, i\in \mathbb{N}$$. If $$Y$$ is OGS, then the array $$X_i$$ is said to belong to the domain of (geometric stable) attraction of $$Y$$. First the paper collects some general properties of OGS and operator stable laws which are partially known in more general situations as long as strict stability is assumed, i.e. if the shift terms $$b_p$$ vanish. Section 4 is concerned in particular with geometric stable laws, i.e. with scalar normalisations $$A_n =a_n I$$, and their marginals and densities; in particular laws with Linnik and Laplace marginals. These distributions turn out to have interesting applications in mathematical finance. The proofs of the main results and some auxiliary results are found in an Appendix.

##### MSC:
 60E07 Infinitely divisible distributions; stable distributions 60F05 Central limit and other weak theorems 60G50 Sums of independent random variables; random walks 62H05 Characterization and structure theory for multivariate probability distributions; copulas
Full Text:
##### References:
 [1] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1965), Dover Publications New York · Zbl 0515.33001 [2] Brorsen, W.B.; Yang, S.R., Maximum likelihood estimates of symmetric stable distribution parameters, Comm. statist. simulation comput, 19, 4, 1459-1464, (1990) · Zbl 0850.62248 [3] S. Cambanis, A. Taraporevala, Infinitely divisible distributions with stable marginals, preprint, 1994. [4] V. Feller, Introduction to the Theory of Probability and its Applications, Vol. 2, 2nd Edition, Wiley, New York, 1971. · Zbl 0219.60003 [5] Hazod, W., On the limit behavior of products of a random number of group valued random variables, Theory probab. appl, 39, 374-394, (1994), (English version: Theory Probab. Appl. 32/2 (1995) 249-263) · Zbl 0834.60010 [6] Hazod, W., On geometric convolutions of distributions of group-valued random variables, (), 167-181 · Zbl 0909.60013 [7] Hazod, W.; Khokhlov, Yu.S., On Szasz’s compactness theorem and applications to geometric stability on groups, Probab. math. statist, 16, 143-156, (1996) · Zbl 0856.60011 [8] Hazod, W.; Siebert, E., Stable probability measures on Euclidean spaces and on locally compact groups, (2001), Kluwer Academic Publishers Dordrecht · Zbl 1002.60002 [9] Huff, B.V., The strict subordination of differential processes, Sankhya ser. A, 31, 403-412, (1969) · Zbl 0211.48102 [10] Jurek, Z.; Mason, J.D., Operator-limit distributions in probability theory, (1993), Wiley New York · Zbl 0850.60003 [11] Kalashnikov, V., Geometric sums: bounds for rare events with applications, (1997), Kluwer Academic Publishers Dordrecht · Zbl 0881.60043 [12] Klebanov, L.B.; Maniya, G.M.; Melamed, I.A., A problem of zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables, Theory probab. appl, 29, 791-794, (1984) · Zbl 0579.60016 [13] Klebanov, L.B.; Rachev, S.T., Sums of random number of random variables and their approximations with ν-accompanying infinitely divisible laws, Serdica math. J, 22, 471-496, (1996) · Zbl 0939.60003 [14] Kotz, S.; Kozubowski, T.J.; Podgórski, K., The Laplace distribution and generalizations: A revisit with applications to communications, economics, engineering, and finance, (2001), Birkhäuser Boston · Zbl 0977.62003 [15] Kozubowski, T.J., The inner characterization of geometric stable laws, Statist. decisions, 12, 307-321, (1994) · Zbl 0923.60020 [16] Kozubowski, T.J., Characterization of multivariate geometric stable distributions, Statist. decisions, 15, 397-416, (1997) · Zbl 0891.60019 [17] T.J. Kozubowski, A general functional relation between a random variable and its length biased counterpart, Technical Report no. 52, Department of Mathematics, University of Nevada at Reno, 2001. [18] T.J. Kozubowski, M.M. Meerschaert, H.P. Scheffler, The operator ν-stable laws, Publ. Math. Debrecen, 2003, in press. · Zbl 1059.60019 [19] Kozubowski, T.J.; Panorska, A.K., On moments and tail behavior of ν-stable random variables, Statist. probab. lett, 29, 307-315, (1996) · Zbl 0905.60017 [20] Kozubowski, T.J.; Panorska, A.K., Weak limits for multivariate random sums, J. multivariate anal, 67, 398-413, (1998) · Zbl 0928.60013 [21] Kozubowski, T.J.; Panorska, A.K., Multivariate geometric stable distributions in financial applications, Math. comput. modelling, 29, 83-92, (1999) · Zbl 1098.62587 [22] Kozubowski, T.J.; Podgórski, K., Asymmetric Laplace distributions, Math. sci, 25, 37-46, (2000) · Zbl 0961.60026 [23] Kozubowski, T.J.; Podgórski, K., A multivariate and asymmetric generalization of Laplace distribution, Comput. statist, 15, 531-540, (2000) · Zbl 1035.60010 [24] Kozubowski, T.J.; Podgórski, K., Asymmetric Laplace laws and modeling financial data, Math. comput. modelling, 34, 1003-1021, (2001) · Zbl 1002.60012 [25] Kozubowski, T.J.; Podgórski, K.; Samorodnitsky, G., Tails of Lévy measure of geometric stable random variables, Extremes, 1, 3, 367-378, (1998) · Zbl 0930.60008 [26] Kozubowski, T.J.; Rachev, S.T., The theory of geometric stable distributions and its use in modeling financial data, European J. oper. res, 74, 310-324, (1994) · Zbl 0803.90008 [27] Kozubowski, T.J.; Rachev, S.T., Multivariate geometric stable laws, J. comput. anal. appl, 1, 4, 349-385, (1999) · Zbl 0964.62039 [28] Madan, D.B.; Carr, P.P.; Chang, E.C., The variance gamma process and option pricing, European finance rev, 2, 79-105, (1998) · Zbl 0937.91052 [29] McCulloch, J.H., Linear regression with stable disturbances, (), 359-376 · Zbl 1042.62574 [30] Meerschaert, M., Regular variation in $$R\^{}\{k\}$$ and vector-normed domains of attraction, Statist. probab. lett, 11, 287-289, (1991) · Zbl 0728.60024 [31] Meerschaert, M.M.; Scheffler, H.P., Limit distributions for sums of independent random vectors, (2001), Wiley New York · Zbl 0990.60003 [32] Meerschaert, M.M.; Scheffler, H.P., Portfolio modeling with heavy tailed random vectors, (), 595-640 [33] Mittnik, S.; Rachev, S.T., Alternative multivariate stable distributions and their applications to financial modelling, (), 107-119 [34] Mittnik, S.; Rachev, S.T., Modeling asset returns with alternative stable distributions, Econometric rev, 12, 3, 261-330, (1993) · Zbl 0801.62096 [35] Mittnik, S.; Rachev, S.T., Reply to comments on “modeling asset returns with alternative stable distributions” and some extensions, Econometric rev, 12, 3, 347-389, (1993) · Zbl 0801.62096 [36] Mittnik, S.; Rachev, S.T.; Doganoglu, T.; Chenyao, D., Maximum likelihood estimation of stable Paretian models, Math. comput. modelling, 29, 275-293, (1999) · Zbl 1098.62520 [37] Mittnik, S.; Rachev, S.T.; Rüschendorf, L., Test of association between multivariate stable vectors, Math. comput. modelling, 29, 181-195, (1999) · Zbl 1098.62534 [38] Nolan, J.P., Maximum likelihood estimation and diagnostics for stable distributions, (), 379-400 · Zbl 0971.62008 [39] Rachev, S.T.; Mittnik, S., Stable Paretian models in finance, (2000), Wiley Chichester · Zbl 0972.91060 [40] Rachev, S.T.; Samorodnitsky, G., Geometric stable distributions in Banach spaces, J. theoret. probab, 7, 2, 351-373, (1994) · Zbl 0804.60014 [41] Rachev, S.T.; SenGupta, A., Geometric stable distributions and laplace – weibull mixtures, Statist. decisions, 10, 251-271, (1992) · Zbl 0762.60012 [42] Rachev, S.T.; SenGupta, A., Laplace – weibull mixtures for modeling price changes, Management sci, 39, 8, 1029-1038, (1993) · Zbl 0785.90024 [43] Resnick, S.; Greenwood, P., A bivariate stable characterization and domains of attraction, J. multivariate anal, 9, 206-221, (1979) · Zbl 0409.62038 [44] Samorodnitsky, G.; Taqqu, M., Stable non-Gaussian random processes, (1994), Chapman & Hall New York · Zbl 0925.60027 [45] Sato, K.I., Lévy processes and infinitely divisible distributions, (1999), Cambridge University Press Cambridge, MA [46] Szasz, D., On classes of limit distributions for sums of a random number of identically distributed independent random variables, Theory. probab. appl, 27, 401-415, (1972) · Zbl 0277.60016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.