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Volumn 1, Issue 2, 1999, Pages 177-217

Univariate Geometric Stable Laws

Author keywords

Geometric compound; Heavy tail modeling; Linnik distribution; Mittag Leffler law; Random summation

Indexed keywords


EID: 0001057321     PISSN: 15211398     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022629726024     Document Type: Article
Times cited : (72)

References (66)
  • 1
    • 38249012701 scopus 로고
    • A multivariate Linnik distribution
    • D. N. Anderson, A multivariate Linnik distribution, Stat. Probab. Lett. 14, 333-336 (1992).
    • (1992) Stat. Probab. Lett. , vol.14 , pp. 333-336
    • Anderson, D.N.1
  • 2
    • 21144480304 scopus 로고
    • Linnik distributions and processes
    • D. N. Anderson and B. C. Arnold, Linnik distributions and processes, J. Appl. Probab. 30, 330-340 (1993).
    • (1993) J. Appl. Probab. , vol.30 , pp. 330-340
    • Anderson, D.N.1    Arnold, B.C.2
  • 3
    • 0011424873 scopus 로고
    • Some characterizations of the exponential distribution by geometric compounding
    • B. C. Arnold, Some characterizations of the exponential distribution by geometric compounding, SIAM J. Appl. Math. 24, 242-244 (1973).
    • (1973) SIAM J. Appl. Math. , vol.24 , pp. 242-244
    • Arnold, B.C.1
  • 4
    • 0004044694 scopus 로고    scopus 로고
    • University Press, Cambridge
    • J. Bertoin, Lévy Processes, University Press, Cambridge, 1996.
    • (1996) Lévy Processes
    • Bertoin, J.1
  • 5
    • 0030376633 scopus 로고    scopus 로고
    • Composition semigroups and random stability
    • J. Bunge, Composition semigroups and random stability, Ann. Probab. 24, 1476-1489 (1996).
    • (1996) Ann. Probab. , vol.24 , pp. 1476-1489
    • Bunge, J.1
  • 7
    • 38249021003 scopus 로고
    • A note on Linnik distribution
    • L. Devroye, A note on Linnik distribution, Stat. Probab. Lett. 9, 305-306 (1990).
    • (1990) Stat. Probab. Lett. , vol.9 , pp. 305-306
    • Devroye, L.1
  • 8
    • 0346665368 scopus 로고
    • On semi-α-Laplace distributions
    • G. Divanji, On semi-α-Laplace distributions, J. Indian Stat. Assoc. 26, 31-38 (1988).
    • (1988) J. Indian Stat. Assoc. , vol.26 , pp. 31-38
    • Divanji, G.1
  • 12
    • 0040708384 scopus 로고
    • A generalization of the results of Pillai
    • Y. Fuita, A generalization of the results of Pillai, Ann. Inst. Stat. Math. 45, 361-365 (1993).
    • (1993) Ann. Inst. Stat. Math. , vol.45 , pp. 361-365
    • Fuita, Y.1
  • 13
    • 0040114652 scopus 로고
    • Limit theorems for sums of random number of positive independent random variables
    • B. V. Gnedenko, Limit theorems for sums of random number of positive independent random variables, Proc. 6th Berkeley Symp. Math. Stat. Probab. 2, 537-549 (1970).
    • (1970) Proc. 6th Berkeley Symp. Math. Stat. Probab. , vol.2 , pp. 537-549
    • Gnedenko, B.V.1
  • 14
    • 0347295765 scopus 로고
    • Several remarks on a paper by I. N. Kovalenko
    • B. V. Gnedenko and B. Fraier, Several remarks on a paper by I. N. Kovalenko, Litovsk. Math. Sb. 9(1), 181-187 (1969).
    • (1969) Litovsk. Math. Sb. , vol.9 , Issue.1 , pp. 181-187
    • Gnedenko, B.V.1    Fraier, B.2
  • 15
    • 0038930480 scopus 로고
    • A characteristic property of one class of limit distributions
    • B. V. Gnedenko and S. Janjic, A characteristic property of one class of limit distributions, Math. Nachr. 113, 145-149 (1983).
    • (1983) Math. Nachr. , vol.113 , pp. 145-149
    • Gnedenko, B.V.1    Janjic, S.2
  • 16
    • 0039220466 scopus 로고
    • The strict subordination of differential processes
    • B. W. Huff, The strict subordination of differential processes, Sankhya Ser. A 31, 403-412 (1969).
    • (1969) Sankhya Ser. A , vol.31 , pp. 403-412
    • Huff, B.W.1
  • 17
    • 0347295767 scopus 로고
    • Preservation of type under mixing
    • S. Janković, Preservation of type under mixing, Teor. Veroyatn. Ee Primen. 37, 594-599 (1992).
    • (1992) Teor. Veroyatn. Ee Primen. , vol.37 , pp. 594-599
    • Janković, S.1
  • 18
    • 21144480228 scopus 로고
    • The first-order autoregressive Mittag-Leffler process
    • K. Jayakumar and R. N. Pillai, The first-order autoregressive Mittag-Leffler process, J. Appl. Probab. 30, 462-466 (1993).
    • (1993) J. Appl. Probab. , vol.30 , pp. 462-466
    • Jayakumar, K.1    Pillai, R.N.2
  • 19
    • 0042805532 scopus 로고
    • Characterization of Distributions by the Method of Intensively Monotone Operators
    • Springer, Berlin
    • A. V. Kakosyan, L. B. Klebanov, and I. A. Melamed, Characterization of Distributions by the Method of Intensively Monotone Operators, Lecture Notes in Math., 1088, Springer, Berlin, 1984.
    • (1984) Lecture Notes in Math. , vol.1088
    • Kakosyan, A.V.1    Klebanov, L.B.2    Melamed, I.A.3
  • 21
    • 0010081441 scopus 로고
    • On doubly stochastic Poison processes
    • J. F. C. Kingman, On doubly stochastic Poison processes, Proc. Cambridge Philos. Soc. 60, 923-930 (1964).
    • (1964) Proc. Cambridge Philos. Soc. , vol.60 , pp. 923-930
    • Kingman, J.F.C.1
  • 22
    • 0040999038 scopus 로고    scopus 로고
    • Sums of random number of random variables and their approximations with v-accompanying infinitely divisible laws
    • L. B. Klebanov and S. T. Rachev, Sums of random number of random variables and their approximations with v-accompanying infinitely divisible laws, Serdica Math. J. 22, 471-496 (1996).
    • (1996) Serdica Math. J. , vol.22 , pp. 471-496
    • Klebanov, L.B.1    Rachev, S.T.2
  • 23
    • 0001479094 scopus 로고
    • A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables
    • L. B. Klebanov, G. M. Maniya, and I. A. Melamed, 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).
    • (1984) Theory Probab. Appl. , vol.29 , pp. 791-794
    • Klebanov, L.B.1    Maniya, G.M.2    Melamed, I.A.3
  • 25
    • 0000140206 scopus 로고
    • Analytic and asymptotic properties of Linnik's probability densities, I and II
    • S. Kotz, I. V. Ostrovskii, and A. Hayfavi, Analytic and asymptotic properties of Linnik's probability densities, I and II, J. Math. Anal. Appl. 193, 353-371, 497-521 (1995).
    • (1995) J. Math. Anal. Appl. , vol.193 , pp. 353-371
    • Kotz, S.1    Ostrovskii, I.V.2    Hayfavi, A.3
  • 26
    • 0029685411 scopus 로고    scopus 로고
    • A mixture representation of the Linnik distribution
    • S. Kotz and I. V. Ostrovskii, A mixture representation of the Linnik distribution, Stat. Probab. Lett. 26, 61-64 (1996).
    • (1996) Stat. Probab. Lett. , vol.26 , pp. 61-64
    • Kotz, S.1    Ostrovskii, I.V.2
  • 27
    • 84893895320 scopus 로고
    • Regression-type estimation of the parameters of stable laws
    • I. A. Koutrouvelis, Regression-type estimation of the parameters of stable laws, J. Amer. Stat. Assoc. 75, 918-928 (1980).
    • (1980) J. Amer. Stat. Assoc. , vol.75 , pp. 918-928
    • Koutrouvelis, I.A.1
  • 29
    • 0348109965 scopus 로고
    • Estimation of the parameters of geometric stable laws
    • Department of Statistics and Applied Probability, University of California, Santa Barbara
    • T. J. Kozubowski, Estimation of the parameters of geometric stable laws, Technical Report No. 253, Department of Statistics and Applied Probability, University of California, Santa Barbara, 1993; to appear in Math. Comput. Modelling, 1998.
    • (1993) Technical Report No. 253
    • Kozubowski, T.J.1
  • 30
    • 0347925965 scopus 로고    scopus 로고
    • T. J. Kozubowski, Estimation of the parameters of geometric stable laws, Technical Report No. 253, Department of Statistics and Applied Probability, University of California, Santa Barbara, 1993; to appear in Math. Comput. Modelling, 1998.
    • (1998) Math. Comput. Modelling
  • 31
    • 0002956688 scopus 로고
    • Representation and properties of geometric stable laws
    • (G. Anastassiou and S. T. Rachev, eds.), Plenum, New York
    • T. J. Kozubowski, Representation and properties of geometric stable laws, in Approximation, Probability, and Related Fields (G. Anastassiou and S. T. Rachev, eds.), Plenum, New York, 1994, pp. 321-337.
    • (1994) Approximation, Probability, and Related Fields , pp. 321-337
    • Kozubowski, T.J.1
  • 32
    • 0003184997 scopus 로고
    • The inner characterization of geometric stable laws
    • T. J. Kozubowski, The inner characterization of geometric stable laws, Stat. Decisions, 12, 307-321 (1994).
    • (1994) Stat. Decisions , vol.12 , pp. 307-321
    • Kozubowski, T.J.1
  • 34
    • 0032086301 scopus 로고    scopus 로고
    • Mixture representation of Linnik distribution revisited
    • T. J. Kozubowski, Mixture representation of Linnik distribution revisited, Stat. Probab. Lett. 38, 157-160 (1998).
    • (1998) Stat. Probab. Lett. , vol.38 , pp. 157-160
    • Kozubowski, T.J.1
  • 35
    • 0002607918 scopus 로고    scopus 로고
    • Exponential mixture representation of geometric stable distributions
    • to appear
    • T. J. Kozubowski, Exponential mixture representation of geometric stable distributions, Ann. Inst. Statist. Math. (to appear).
    • Ann. Inst. Statist. Math.
    • Kozubowski, T.J.1
  • 38
    • 0028405668 scopus 로고
    • The theory of geometric stable distributions and its use in modeling financial data
    • T. J. Kozubowski and S. T. Rachev, The theory of geometric stable distributions and its use in modeling financial data, Eur. J. Oper. Res. 74, 310-324 (1994).
    • (1994) Eur. J. Oper. Res. , vol.74 , pp. 310-324
    • Kozubowski, T.J.1    Rachev, S.T.2
  • 39
    • 0030590323 scopus 로고    scopus 로고
    • On moments and tail behavior of v-stable random variables
    • T. J. Kozubowski and A. K. Panorska, On moments and tail behavior of v-stable random variables, Stat. Probab. Lett. 29, 307-315 (1996).
    • (1996) Stat. Probab. Lett. , vol.29 , pp. 307-315
    • Kozubowski, T.J.1    Panorska, A.K.2
  • 41
  • 43
    • 84968497918 scopus 로고
    • On a class of unimodal distributions
    • R. G. Laha, On a class of unimodal distributions, Proc. Amer. Math. Soc. 12, 181-184 (1961).
    • (1961) Proc. Amer. Math. Soc. , vol.12 , pp. 181-184
    • Laha, R.G.1
  • 44
    • 0038883234 scopus 로고
    • Characterizations of the Laplace and related distributions via geometric compounding
    • G. D. Lin, Characterizations of the Laplace and related distributions via geometric compounding, Sankhya Ser. A 56, 1-9 (1994).
    • (1994) Sankhya Ser. A , vol.56 , pp. 1-9
    • Lin, G.D.1
  • 45
    • 0002433318 scopus 로고
    • Linear forms and statistical criteria, I, II
    • Ju. V. Linnik, Linear forms and statistical criteria, I, II, Selected Translations in Math. Stat. Probab. 3, 1-90 (1963). [Original paper in Ukrainskii Mat. Zh. 5, 207-290 (1953).]
    • (1963) Math. Stat. Probab. , vol.3 , pp. 1-90
    • Linnik, J.V.1
  • 46
    • 0000549804 scopus 로고
    • Ju. V. Linnik, Linear forms and statistical criteria, I, II, Selected Translations in Math. Stat. Probab. 3, 1-90 (1963). [Original paper in Ukrainskii Mat. Zh. 5, 207-290 (1953).]
    • (1953) Ukrainskii Mat. Zh. , vol.5 , pp. 207-290
  • 47
    • 0000642461 scopus 로고
    • On the distribution of stock price differences
    • B. Mandelbrot and H. M. Taylor, On the distribution of stock price differences, Oper. Res. 15, 1057-1062 (1967).
    • (1967) Oper. Res. , vol.15 , pp. 1057-1062
    • Mandelbrot, B.1    Taylor, H.M.2
  • 48
    • 0012046867 scopus 로고
    • Random sum characterization
    • R. K. Milne and G. F. Yeo, Random sum characterization, Math. Scientist. 14, 120-126 (1989).
    • (1989) Math. Scientist. , vol.14 , pp. 120-126
    • Milne, R.K.1    Yeo, G.F.2
  • 49
    • 0001844051 scopus 로고
    • Alternative multivariate stable distributions and their applications to financial modelling
    • (S. Cambanis et al., eds.), Birkhaüser, Boston
    • S. Mittnik and S. T. Rachev, Alternative multivariate stable distributions and their applications to financial modelling, in Stable Processes and Related Topics (S. Cambanis et al., eds.), Birkhaüser, Boston, 1991, pp. 107-119.
    • (1991) Stable Processes and Related Topics , pp. 107-119
    • Mittnik, S.1    Rachev, S.T.2
  • 50
    • 84958363567 scopus 로고
    • Modeling asset returns with alternative stable distributions
    • S. Mittnik and S. T. Rachev, Modeling asset returns with alternative stable distributions, Econometric Rev. 12, 261-330 (1993).
    • (1993) Econometric Rev. , vol.12 , pp. 261-330
    • Mittnik, S.1    Rachev, S.T.2
  • 51
    • 0347925963 scopus 로고
    • On geometrically infinitely divisible laws and geometric domains of attraction
    • M. R. Mohan, R. Vasudeva, and H. V. Hebbar, On geometrically infinitely divisible laws and geometric domains of attraction, Sankhya Ser. A 55, 171-179 (1993).
    • (1993) Sankhya Ser. A , vol.55 , pp. 171-179
    • Mohan, M.R.1    Vasudeva, R.2    Hebbar, H.V.3
  • 53
    • 84995107546 scopus 로고
    • A characterization of gamma mixtures of stable laws motivated by limit theorems
    • A. G. Pakes, A characterization of gamma mixtures of stable laws motivated by limit theorems, Stat. Neerl. 2-3, 209-218 (1992).
    • (1992) Stat. Neerl. , vol.2-3 , pp. 209-218
    • Pakes, A.G.1
  • 54
    • 0003191865 scopus 로고
    • Semi-α-Laplace distributions
    • R. N. Pillai, Semi-α-Laplace distributions, Commun. Stat. Theory Methods 14, 991-1000 (1985).
    • (1985) Commun. Stat. Theory Methods , vol.14 , pp. 991-1000
    • Pillai, R.N.1
  • 55
    • 0000495328 scopus 로고
    • On Mittag-Leffler functions and related distributions
    • R. N. Pillai, On Mittag-Leffler functions and related distributions, Ann. Inst. Stat. Math. 42, 157-161 (1990).
    • (1990) Ann. Inst. Stat. Math. , vol.42 , pp. 157-161
    • Pillai, R.N.1
  • 56
    • 0003260703 scopus 로고
    • A certain class of distributions under normal attraction
    • R. N. Pillai, and Sabu George, A certain class of distributions under normal attraction, Proc. VIth Annu. Conf. ISPS. pp. 107-112 (1984).
    • (1984) Proc. VIth Annu. Conf. ISPS. , pp. 107-112
    • Pillai, R.N.1    George, S.2
  • 57
    • 0011671091 scopus 로고
    • Distributions with complete monotone derivative and geometric infinite divisibility
    • R. N. Pillai and E. Sandhya, Distributions with complete monotone derivative and geometric infinite divisibility, Adv. Appl. Probab. 22, 751-754 (1990).
    • (1990) Adv. Appl. Probab. , vol.22 , pp. 751-754
    • Pillai, R.N.1    Sandhya, E.2
  • 58
    • 0001697927 scopus 로고
    • Analytic and asymptotic properties of multivariate Linnik's distribution
    • I. V. Ostrovskii, Analytic and asymptotic properties of multivariate Linnik's distribution, Math. Phys. Anal. Geom. 2, 436-455 (1995).
    • (1995) Math. Phys. Anal. Geom. , vol.2 , pp. 436-455
    • Ostrovskii, I.V.1
  • 59
    • 0001744721 scopus 로고
    • Geometric stable distributions in Banach spaces
    • S. T. Rachev and G. Samorodnitsky, Geometric stable distributions in Banach spaces, J. Theoret. Probab. 7, 351-373 (1994).
    • (1994) J. Theoret. Probab. , vol.7 , pp. 351-373
    • Rachev, S.T.1    Samorodnitsky, G.2
  • 60
    • 9844239371 scopus 로고    scopus 로고
    • On geometric stable laws, a related property of stable processes, and stable densities
    • B. Ramachandran, On geometric stable laws, a related property of stable processes, and stable densities, Ann. Inst. Stat. Math. 49, 299-313 (1997).
    • (1997) Ann. Inst. Stat. Math. , vol.49 , pp. 299-313
    • Ramachandran, B.1
  • 63
    • 0030585552 scopus 로고    scopus 로고
    • On the Chambers-Mallows-Stuck method for simulating skewed stable random variables
    • R. Weron, On the Chambers-Mallows-Stuck method for simulating skewed stable random variables, Stat. Probab. Lett. 28, 165-171 (1996).
    • (1996) Stat. Probab. Lett. , vol.28 , pp. 165-171
    • Weron, R.1
  • 64
    • 0030264775 scopus 로고    scopus 로고
    • On the Cole-Cole relaxation function and related Mittag-Leffler distributions
    • K. Weron and M. Kotulski, On the Cole-Cole relaxation function and related Mittag-Leffler distributions, Physica A, 232, 180-188 (1996).
    • (1996) Physica A , vol.232 , pp. 180-188
    • Weron, K.1    Kotulski, M.2
  • 65
    • 0000472011 scopus 로고
    • Unimodality of infinitely divisible distribution functions of class L
    • M. Yamazato, Unimodality of infinitely divisible distribution functions of class L, Ann. Probab. 6, 523-531 (1978).
    • (1978) Ann. Probab. , vol.6 , pp. 523-531
    • Yamazato, M.1
  • 66
    • 0003588120 scopus 로고
    • of Translations of Mathematical Monographs, American Mathematical Society
    • V. M. Zolotarev, One-Dimensional Stable Distributions, Vol. 65 of Translations of Mathematical Monographs, American Mathematical Society, 1986.
    • (1986) One-Dimensional Stable Distributions , vol.65
    • Zolotarev, V.M.1


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