메뉴 건너뛰기




Volumn 25, Issue 2, 1997, Pages 855-900

The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator

Author keywords

Local time; Poisson point process; Ranked lengths of excursions; Semistable Markov process; Zero set

Indexed keywords


EID: 0031534984     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1024404422     Document Type: Article
Times cited : (901)

References (76)
  • 2
    • 0002478027 scopus 로고
    • On mixing and stability of limit theorems
    • ALDOUS, D. J. and EAGLESON, G. K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6 325-331.
    • (1978) Ann. Probab. , vol.6 , pp. 325-331
    • Aldous, D.J.1    Eagleson, G.K.2
  • 3
    • 84990663423 scopus 로고
    • Brownian bridge asymptotics for random mappings
    • ALDOUS, D. and PITMAN, J. (1994). Brownian bridge asymptotics for random mappings. Random Structures Algorithms 5 487-512.
    • (1994) Random Structures Algorithms , vol.5 , pp. 487-512
    • Aldous, D.1    Pitman, J.2
  • 4
    • 0001616352 scopus 로고
    • The extreme terms of a sample and their role in the sum of independent variables
    • AROV, D. and BOBROV, A. (1960). The extreme terms of a sample and their role in the sum of independent variables. Theory Probab. Appl. 5 377-396.
    • (1960) Theory Probab. Appl. , vol.5 , pp. 377-396
    • Arov, D.1    Bobrov, A.2
  • 6
    • 0001236422 scopus 로고
    • On the distribution of large prime factors
    • BILLINGSLEY, P. (1972). On the distribution of large prime factors. Period. Math. Hungar. 2 283-289.
    • (1972) Period. Math. Hungar. , vol.2 , pp. 283-289
    • Billingsley, P.1
  • 8
    • 0002617436 scopus 로고
    • Ferguson distributions via Pólya urn schemes
    • BLACKWELL, D. and MACQUEEN, J. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353-355.
    • (1973) Ann. Statist. , vol.1 , pp. 353-355
    • Blackwell, D.1    Macqueen, J.2
  • 10
    • 84968502875 scopus 로고
    • On the application of the Borel-Cantelli Lemma
    • CHUNG, K. and ERDÖS, P. (1952). On the application of the Borel-Cantelli Lemma. Trans. Amer. Math. Soc, 72 179-186.
    • (1952) Trans. Amer. Math. Soc , vol.72 , pp. 179-186
    • Chung, K.1    Erdös, P.2
  • 12
    • 0009253021 scopus 로고
    • The influence of the maximum term in the addition of independent random variables
    • DARLING, D. (1952). The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73 95-107.
    • (1952) Trans. Amer. Math. Soc. , vol.73 , pp. 95-107
    • Darling, D.1
  • 13
    • 0002100612 scopus 로고
    • On the frequency of numbers containing prime factors of a certain relative magnitude
    • DICKMAN, K. (1930). On the frequency of numbers containing prime factors of a certain relative magnitude. Arkiv. for Matematik Astronomi och Fysik 22 1-14.
    • (1930) Arkiv. for Matematik Astronomi Och Fysik , vol.22 , pp. 1-14
    • Dickman, K.1
  • 14
    • 0022798579 scopus 로고
    • Partition structures, Pólya urns, the Ewens sampling formula, and the ages of alleles
    • DONNELLY, P. (1986). Partition structures, Pólya urns, the Ewens sampling formula, and the ages of alleles. Theoret. Population Biol. 30 271-288.
    • (1986) Theoret. Population Biol. , vol.30 , pp. 271-288
    • Donnelly, P.1
  • 15
    • 84963060268 scopus 로고
    • On the asymptotic distribution of large prime factors
    • DONNELLY, P. and GRIMMETT, G. (1993). On the asymptotic distribution of large prime factors. J. London Math. Soc. (2) 47 395-404.
    • (1993) J. London Math. Soc. , vol.47 , Issue.2 , pp. 395-404
    • Donnelly, P.1    Grimmett, G.2
  • 16
    • 38249024666 scopus 로고
    • Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex
    • DONNELLY, P. and JOYCE, P. (1989). Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex. Stochastic Process. Appl. 31 89-103.
    • (1989) Stochastic Process. Appl. , vol.31 , pp. 89-103
    • Donnelly, P.1    Joyce, P.2
  • 18
  • 19
    • 0015310384 scopus 로고
    • The sampling theory of selectively neutral alleles
    • EWENS, W. (1972). The sampling theory of selectively neutral alleles. Theoret. Population Biol. 3 87-112.
    • (1972) Theoret. Population Biol. , vol.3 , pp. 87-112
    • Ewens, W.1
  • 20
    • 0002357984 scopus 로고
    • Population genetics theory-The past and the future
    • S. Lessard, ed., Univ. Montreal Press
    • EWENS, W. (1988). Population genetics theory-the past and the future. In Mathematical and Statistical Problems in Evolution (S. Lessard, ed.), Univ. Montreal Press.
    • (1988) Mathematical and Statistical Problems in Evolution
    • Ewens, W.1
  • 21
    • 0010953182 scopus 로고
    • The Ewens sampling formula
    • N. S. Johnson, S. Kotz and N. Balakrishnan, eds.. Wiley, New York. To appear
    • EWENS, W. and TAVARÉ, S. (1995). The Ewens sampling formula. In Multivariate Discrete Distributions (N. S. Johnson, S. Kotz and N. Balakrishnan, eds.). Wiley, New York. To appear.
    • (1995) Multivariate Discrete Distributions
    • Ewens, W.1    Tavaré, S.2
  • 22
    • 0001120413 scopus 로고
    • A Bayesian analysis of some nonparametric problems
    • FERGUSON, T. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230.
    • (1973) Ann. Statist. , vol.1 , pp. 209-230
    • Ferguson, T.1
  • 23
    • 0039615411 scopus 로고
    • The asymptotic distribution of the number of zero free intervals of a stable process
    • GETOOR, R. (1963). The asymptotic distribution of the number of zero free intervals of a stable process. Trans. Amer. Math. Soc. 106 127-138.
    • (1963) Trans. Amer. Math. Soc. , vol.106 , pp. 127-138
    • Getoor, R.1
  • 24
    • 0001975329 scopus 로고
    • On the field of combinatory analysis
    • GONČHAROV, V. (1962). On the field of combinatory analysis. Amer. Math. Soc. Transl. 19 1-46.
    • (1962) Amer. Math. Soc. Transl. , vol.19 , pp. 1-46
    • Gončharov, V.1
  • 25
    • 0010523756 scopus 로고
    • Exact sampling distributions from the infinite neutral alleles models
    • GRIFFITHS, R. C. (1979). Exact sampling distributions from the infinite neutral alleles models. Adv. in Appl. Probab. 11 326-354.
    • (1979) Adv. in Appl. Probab. , vol.11 , pp. 326-354
    • Griffiths, R.C.1
  • 26
    • 0040801507 scopus 로고
    • On the distribution of points in a Poisson Dirichlet process
    • GRIFFITHS, R. C. (1988). On the distribution of points in a Poisson Dirichlet process. J. Appl. Probab. 25 336-345.
    • (1988) J. Appl. Probab. , vol.25 , pp. 336-345
    • Griffiths, R.C.1
  • 27
    • 84990648257 scopus 로고
    • Order statistics for decomposable combinatorial structures
    • HANSEN, J. (1994). Order statistics for decomposable combinatorial structures. Random Structures Algorithms 5 517-533.
    • (1994) Random Structures Algorithms , vol.5 , pp. 517-533
    • Hansen, J.1
  • 28
    • 0021634009 scopus 로고
    • Pólya-like urns and the Ewens sampling formula
    • HOPPE, F. M. (1984). Pólya-like urns and the Ewens sampling formula. J. Math. Biol. 20 91-94.
    • (1984) J. Math. Biol. , vol.20 , pp. 91-94
    • Hoppe, F.M.1
  • 29
    • 0040207172 scopus 로고
    • Size-biased filtering of Poisson-Dirichlet samples with an application to partition structures in genetics
    • HOPPE, F. M. (1986). Size-biased filtering of Poisson-Dirichlet samples with an application to partition structures in genetics. J. Appl. Probab. 23 1008-1012.
    • (1986) J. Appl. Probab. , vol.23 , pp. 1008-1012
    • Hoppe, F.M.1
  • 30
    • 0023069772 scopus 로고
    • The sampling theory of neutral alleles and an urn model in population genetics
    • HOPPE, F. M. (1987). The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25 123-159.
    • (1987) J. Math. Biol. , vol.25 , pp. 123-159
    • Hoppe, F.M.1
  • 31
    • 0039023126 scopus 로고
    • A note on the arc-sine law and Markov random sets
    • HOROWITZ, J. (1971). A note on the arc-sine law and Markov random sets. Ann. Math. Statist. 42 1068-1074.
    • (1971) Ann. Math. Statist. , vol.42 , pp. 1068-1074
    • Horowitz, J.1
  • 32
    • 0041147221 scopus 로고
    • Semilinear Markov processes, subordinators and renewal theory
    • HOROWITZ, J. (1972). Semilinear Markov processes, subordinators and renewal theory. Z. Wahrsch. Verw. Gebiete 24 167-193.
    • (1972) Z. Wahrsch. Verw. Gebiete , vol.24 , pp. 167-193
    • Horowitz, J.1
  • 34
    • 0001688105 scopus 로고
    • On a constant arising in the theory of symmetric groups and on Poisson-Dirichlet measures
    • IGNATOV, T. (1982). On a constant arising in the theory of symmetric groups and on Poisson-Dirichlet measures. Theory Probab. Appl. 27 136-147.
    • (1982) Theory Probab. Appl. , vol.27 , pp. 136-147
    • Ignatov, T.1
  • 35
    • 0007373304 scopus 로고
    • Canonical representations and convergence criteria for processes with interchangeable increments
    • KALLENBERG, O. (1973). Canonical representations and convergence criteria for processes with interchangeable increments. Z. Wahrsch. Verw. Gebiete 27 23-36.
    • (1973) Z. Wahrsch. Verw. Gebiete , vol.27 , pp. 23-36
    • Kallenberg, O.1
  • 39
    • 0039023133 scopus 로고
    • On the duration of the longest excursion
    • E. Çinlar, K. Chung and R. Getoor, eds. . Birkhäuser, Basel
    • KNIGHT, F. (1985). On the duration of the longest excursion. In Seminar on Stochastic Processes (E. Çinlar, K. Chung and R. Getoor, eds.) 117-148. Birkhäuser, Basel.
    • (1985) Seminar on Stochastic Processes , pp. 117-148
    • Knight, F.1
  • 40
    • 0000991636 scopus 로고
    • Exponential families, extreme point models and minimal space-time invariant functions for stochastic processes with stationary and independent increments
    • KÜCHLER, U. and LAURITZEN, S. L. (1989). Exponential families, extreme point models and minimal space-time invariant functions for stochastic processes with stationary and independent increments. Scand. J. Statist. 16 237-261.
    • (1989) Scand. J. Statist. , vol.16 , pp. 237-261
    • Küchler, U.1    Lauritzen, S.L.2
  • 41
    • 84968497856 scopus 로고
    • An occupation time theorem for a class of stochastic processes
    • LAMPERTI, J. (1958). An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc. 88 380-387.
    • (1958) Trans. Amer. Math. Soc. , vol.88 , pp. 380-387
    • Lamperti, J.1
  • 42
    • 84968490634 scopus 로고
    • A contribution to renewal theory
    • LAMPERTI, J. (1961). A contribution to renewal theory. Proc. Amer. Math. Soc. 12 724-731.
    • (1961) Proc. Amer. Math. Soc. , vol.12 , pp. 724-731
    • Lamperti, J.1
  • 43
    • 0007237919 scopus 로고
    • An invariance principle in renewal theory
    • LAMPERTI, J. (1962). An invariance principle in renewal theory. Ann. Math. Statist. 33 685-696.
    • (1962) Ann. Math. Statist. , vol.33 , pp. 685-696
    • Lamperti, J.1
  • 44
  • 45
    • 0000994389 scopus 로고
    • Convergence to a stable distribution via order statistics
    • LEPAGE, R., WOODROOFE, M. and ZINN, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624-632.
    • (1981) Ann. Probab. , vol.9 , pp. 624-632
    • LePage, R.1    Woodroofe, M.2    Zinn, J.3
  • 46
    • 0000071308 scopus 로고
    • Sur certains processus stochastiques homogènes
    • LÉVY, P. (1939). Sur certains processus stochastiques homogènes. Compositio Math. 7 283-339.
    • (1939) Compositio Math. , vol.7 , pp. 283-339
    • Lévy, P.1
  • 47
    • 0001504520 scopus 로고
    • A characterization of the gamma distribution
    • LUKACS, E. (1955), A characterization of the gamma distribution. Ann. Math. Statist. 26 319-324.
    • (1955) Ann. Math. Statist. , vol.26 , pp. 319-324
    • Lukacs, E.1
  • 49
    • 0000944515 scopus 로고
    • Symmetric stable processes as traces of degenerate diffusion processes
    • MOLCHANOV, S. A. and OSTROVSKI, E. (1969). Symmetric stable processes as traces of degenerate diffusion processes. Theory Probab. Appl. 14 128-131.
    • (1969) Theory Probab. Appl. , vol.14 , pp. 128-131
    • Molchanov, S.A.1    Ostrovski, E.2
  • 50
    • 38249001768 scopus 로고
    • Order statistics for jumps of normalized subordinators
    • PERMAN, M. (1993). Order statistics for jumps of normalized subordinators. Stochastic Process Appl. 46 267-281.
    • (1993) Stochastic Process Appl. , vol.46 , pp. 267-281
    • Perman, M.1
  • 51
    • 0001902647 scopus 로고
    • Size-biased sampling of Poisson point processes and excursions
    • PERMAN, M., PITMAN, J. and YOR, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 21-39.
    • (1992) Probab. Theory Related Fields , vol.92 , pp. 21-39
    • Perman, M.1    Pitman, J.2    Yor, M.3
  • 54
    • 21844504293 scopus 로고
    • Exchangeable and partially exchangeable random partitions
    • PITMAN, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145-158.
    • (1995) Probab. Theory Related Fields , vol.102 , pp. 145-158
    • Pitman, J.1
  • 56
    • 0001191605 scopus 로고    scopus 로고
    • Random discrete distributions invariant under size-biased permutation
    • PITMAN, J. (1996). Random discrete distributions invariant under size-biased permutation. Adv. in Appl. Probab. 28 525-539.
    • (1996) Adv. in Appl. Probab. , vol.28 , pp. 525-539
    • Pitman, J.1
  • 59
    • 84962993911 scopus 로고
    • Arcsine laws and interval partitions derived from a stable subordinator
    • PITMAN, J. and YOR, M. (1992). Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. (3) 65 326-356.
    • (1992) Proc. London Math. Soc. , vol.65 , Issue.3 , pp. 326-356
    • Pitman, J.1    Yor, M.2
  • 60
    • 0001280730 scopus 로고    scopus 로고
    • On the relative lengths of excursions derived from a stable subordinator
    • To appear
    • PITMAN, J. and YOR, M. (1996). On the relative lengths of excursions derived from a stable subordinator. Séminaire de Probabilités XXXI. To appear.
    • (1996) Séminaire de Probabilités , vol.31
    • Pitman, J.1    Yor, M.2
  • 61
    • 0002095272 scopus 로고
    • Point processes, regular variation and weak convergence
    • RESNICK, S. I. (1986). Point processes, regular variation and weak convergence. Adv. in Appl. Probab. 18 66-138.
    • (1986) Adv. in Appl. Probab. , vol.18 , pp. 66-138
    • Resnick, S.I.1
  • 62
    • 0000037557 scopus 로고
    • The rank of the present excursion
    • SCHEFFER, C. (1995). The rank of the present excursion. Stochastic Process Appl. 55 101-118.
    • (1995) Stochastic Process Appl. , vol.55 , pp. 101-118
    • Scheffer, C.1
  • 63
    • 84968490734 scopus 로고
    • Ordered cycle lengths in a random permutation
    • SHEPP, L. and LLOYD, S. (1966). Ordered cycle lengths in a random permutation. Trans. Amer. Math. Soc. 121 340-357.
    • (1966) Trans. Amer. Math. Soc. , vol.121 , pp. 340-357
    • Shepp, L.1    Lloyd, S.2
  • 64
    • 0001012341 scopus 로고
    • Limit distributions of certain characteristics of random mappings
    • STEPANOV, V. (1969). Limit distributions of certain characteristics of random mappings. Theory Probab. Appl. 14 612-626.
    • (1969) Theory Probab. Appl. , vol.14 , pp. 612-626
    • Stepanov, V.1
  • 65
    • 0010953089 scopus 로고
    • The asymptotic distribution of factorizations of natural numbers into prime divisors
    • VERSHIK, A. M. (1986). The asymptotic distribution of factorizations of natural numbers into prime divisors. Soviet Math. Dokl. 34 57-61.
    • (1986) Soviet Math. Dokl. , vol.34 , pp. 57-61
    • Vershik, A.M.1
  • 66
    • 0002584967 scopus 로고
    • Limit measures arising in the theory of groups, I
    • VERSHIK, A. and SHMIDT, A. (1977). Limit measures arising in the theory of groups, I. Theory Probab. Appl. 22 79-85.
    • (1977) Theory Probab. Appl. , vol.22 , pp. 79-85
    • Vershik, A.1    Shmidt, A.2
  • 67
    • 0013221038 scopus 로고
    • Limit measures arising in the theory of symmetric groups, II
    • VERSHIK, A. and SHMIDT, A. (1978). Limit measures arising in the theory of symmetric groups, II. Theory Probab. Appl. 23 36-49.
    • (1978) Theory Probab. Appl. , vol.23 , pp. 36-49
    • Vershik, A.1    Shmidt, A.2
  • 70
    • 0001565579 scopus 로고
    • On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables
    • VERVAAT, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 750-783.
    • (1979) Adv. in Appl. Probab. , vol.11 , pp. 750-783
    • Vervaat, W.1
  • 71
    • 0000893434 scopus 로고
    • Generalized arc-sine laws for one-dimensional diffusion processes and random walks
    • WATANABE, S. (1995). Generalized arc-sine laws for one-dimensional diffusion processes and random walks. Proc. Sympos. Pure Math. 57 157-172.
    • (1995) Proc. Sympos. Pure Math. , vol.57 , pp. 157-172
    • Watanabe, S.1
  • 72
    • 0001276580 scopus 로고
    • The stationary distribution of the infinitely-many neutral alleles diffusion model
    • WATTERSON, G. A. (1976). The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Probab. 13 639-651.
    • (1976) J. Appl. Probab. , vol.13 , pp. 639-651
    • Watterson, G.A.1
  • 73
  • 74
    • 0007391298 scopus 로고
    • Zero-free intervals of semi-stable Markov processes
    • WENDEL, J. (1964). Zero-free intervals of semi-stable Markov processes. Math. Scand. 14 21-34.
    • (1964) Math. Scand. , vol.14 , pp. 21-34
    • Wendel, J.1
  • 75
    • 0042692390 scopus 로고    scopus 로고
    • The continuum of inductive methods revisited
    • J. Earman and J. Norton, eds.. Univ. Pittsburgh Press/Univ. Konstanz. To appear
    • ZABELL, S. (1996). The continuum of inductive methods revisited. In The Cosmos of Science (J. Earman and J. Norton, eds.). Univ. Pittsburgh Press/Univ. Konstanz. To appear.
    • (1996) The Cosmos of Science
    • Zabell, S.1
  • 76
    • 0009425345 scopus 로고
    • Non-self-averaging effects in sums of random variables, spin glasses, random maps and random walks
    • M. Fannes, C. Maes and A. Verbeure, eds. NATO ASI Series Plenum Press, New York
    • DERRIDA, B. (1994). Non-self-averaging effects in sums of random variables, spin glasses, random maps and random walks. In On Three Levels: Micro-, Meso-, and Macro-Approaches in Physics (M. Fannes, C. Maes and A. Verbeure, eds.) NATO ASI Series 125-137. Plenum Press, New York.
    • (1994) On Three Levels: Micro-, Meso-, and Macro-approaches in Physics , pp. 125-137
    • Derrida, B.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.