Advertisement

Exchangeability and related topics

  • David J. Aldous
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1117)

Notation

"Positive", "increasing" are used in the weak sense

R

set of real numbers

ℤ; ℕ

set of all integers; set of natural numbers

#A

cardinality of set A

1A

indicator function/random variable: 1A(x)=1 for xεA=0 else

δa(·)

probability measure degenerate at a: δa(A)=1A(a)

F ⊂ G a.s.

for each G ε G there exists F ε F such that P(F Δ G)=0

F=G a.s.

F ⊂ G a.s. and G ⊂ F a.s.

F is trivial

F={φ,ω} a.s.

L(X)

distribution of random variable X

σ(X)

σ-field generated by X

Open image in new window

convergence in probability

Open image in new window

convergence in distribution

N(µ,σ2)

Normal distribution

U(0,1)

Uniform distribution on (0,1)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Accardi, L. and Pistone, G. (1982). de Finetti's theorem, sufficiency, and Dobrushin's theory. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.), North-Holland, Amsterdam, 125–156.Google Scholar
  2. Ahmad, R. (1982). On the structure and application of restricted exchangeability. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.), North-Holland, Amsterdam, 157–164.Google Scholar
  3. Aldous, D. J. (1977). Limit theorems for subsequences of arbitrarily-dependent sequences of random variables. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete 40, 59–82.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Aldous, D. J. (1978). Stopping times and tightness. Ann. Probability 6, 335–340.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Aldous, D. J. (1979). Symmetry and independence for arrays of random variables. Preprint.Google Scholar
  6. Aldous, D. J. (1981a). Representations for partially exchangeable arrays of random variables. J. Multivariate Anal. 11, 581–598.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Aldous, D. J. (1981b). Subspaces of L1, via random measures. Trans. Amer. Math. Soc. 267, 445–463.MathSciNetzbMATHGoogle Scholar
  8. Aldous, D. J. (1982a). Partial exchangeability and Open image in new window-topologies. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 23–38.Google Scholar
  9. Aldous, D. J. (1982b). On exchangeability and conditional independence. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 165–170.Google Scholar
  10. Aldous, D. J. and Eagleson, G. K. (1978). On mixing and stability of limit theorems. Ann. Probability 6, 325–331.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Aldous, D. J. and Pitman, J. (1979). On the zero-one law for exchangeable events. Ann. Probability 7, 704–723.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Arnaud, J.-P. (1980). Fonctions sphériques et fonctions définies positives sur l'arbre homogène. C. R. Acad. Sci. A 290, 99–101.MathSciNetzbMATHGoogle Scholar
  13. Bailey, R., Praeger, C., Speed, T. P. and Taylor, D. (1984). Analysis of Variance. Springer-Verlag, to appear.Google Scholar
  14. Barbour, A. D. and Eagleson, G. K. (1983). Poisson approximations for some statistics based on exchangeable trials. Adv. Appl. Prob. 15, 585–600.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Berbee, H. (1981). On covering single points by randomly ordered intervals. Ann. Probability 9, 520–528.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Berkes, I. and Péter, E. (1983). Exchangeable r.v.'s and the subsequence principle. Preprint, Math. Inst. Hungarian Acad. Sci.Google Scholar
  17. Berkes, I. and Rosenthal, H. P. (1983). Almost exchangeable sequences of random variables. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete (to appear).Google Scholar
  18. Berman, S. (1965). Sign-invariant random variables and stochastic processes with sign-invariant increments. Trans. Amer. Math. Soc. 119, 216–243.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.zbMATHGoogle Scholar
  20. Blackwell, D. and Freedman, D. (1964). The tail σ-field of a Markov chain and a theorem of Orey. Ann. Math. Statist. 35, 1921–1925.MathSciNetzbMATHGoogle Scholar
  21. Blum, J. R. (1982). Exchangeability and quasi-exchangeability. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 171–176.Google Scholar
  22. Blum, J., Chernoff, J., Rosenblatt, M., Teicher, H. (1958). Central limit theorems for interchangeable processes. Canad. J. Math. 10, 222–229.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Breiman, L. (1968). Probability. Addison-Wesley, Reading, Mass.zbMATHGoogle Scholar
  24. Brown, G. and Dooley, A. H. (1983). Ergodic measures are of weak product type. Preprint, School of Mathematics, University of New South Wales.Google Scholar
  25. Brown, T. C. (1982). Poisson approximations and exchangeable random variables. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 177–184.Google Scholar
  26. Bru, B., Heinich, H., Lootgitier, J.-C. (1981). Lois de grands nombres pour les variables échangeables. Comptes Rendus Acad. Sci. 293, 485–488.zbMATHGoogle Scholar
  27. Bühlmann, H. (1958). Le problème "limite centrale" pour les variables aléatoires échangeables. C. R. Acad. Sci. Paris 246, 534–536.MathSciNetzbMATHGoogle Scholar
  28. Bühlmann, H. (1960). Austauschbare stochastische Variabeln und ihre Grenzwertsätze. Univ. Calif. Publ. Statist. 3, 1–35.zbMATHGoogle Scholar
  29. Carnal, E. (1980). Indépendence conditionelle permutable. Ann. Inst. Henri Poincaré B 16, 39–47.MathSciNetzbMATHGoogle Scholar
  30. Cartier, P. (1973). Harmonic analysis on trees. In Harmonic Analysis on Homogenous Spaces. Amer. Math. Soc. (Proc. Sympos. Pure Math. 26). Providence.Google Scholar
  31. Chatterji, S. D. (1974). A subsequence principle in probability theory. Bull. Amer. Math. Soc. 80, 495–497.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Choquet, G. (1969). Lectures on Analysis. Benjamin, Reading, Mass.zbMATHGoogle Scholar
  33. Chow, Y. S. and Teicher, H. (1978). Probability Theory. Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  34. Crisma, L. (1982). Quantitative analysis of exchangeability in alternative processes. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 207–216.Google Scholar
  35. Csörgö, M. and Revesz, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York.zbMATHGoogle Scholar
  36. Daboni, L. (1982). Exchangeability and completely monotone functions. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 39–45.Google Scholar
  37. Dacunha-Castelle, D. (1974). Indiscernability and exchangeability in Lp spaces. Seminar on Random Series, Convex Sets and Geometry of Banach Spaces. Aarhus.Google Scholar
  38. Dacunha-Castelle, D. (1982). A survey of exchangeable random variables in normed spaces. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 47–60.Google Scholar
  39. Dacunha-Castelle, D. and Schreiber, M. (1974). Techniques probabilistes pour l'étude de problèmes d'isomorphismes entre espaces de Banach. Annales Inst. Henri Poincaré 10, 229–277.MathSciNetzbMATHGoogle Scholar
  40. Davidson, R. (1974). Construction of line processes. In Stochastic Geometry (E. F. Harding and D. G. Kendall, eds.). Wiley, New York.Google Scholar
  41. Dawid, A. P. (1977a). Invariant distributions and analysis of variance models. Biometrika 64, 291–297.MathSciNetzbMATHCrossRefGoogle Scholar
  42. Dawid, A. P. (1977b). Spherical matrix distributions and a multivariate model. J. Roy. Statist. Soc. B 39, 254–261.MathSciNetzbMATHGoogle Scholar
  43. Dawid, A. P. (1978). Extendibility of spherical matrix distributions. J. Multivariate Anal. 8, 567–572.MathSciNetzbMATHCrossRefGoogle Scholar
  44. Dawid, A. P. (1979). Conditional independence in statistical theory. J. Roy. Statist. Soc. B 41, 1–31.MathSciNetzbMATHGoogle Scholar
  45. Dawid, A. P. (1982). Intersubjective statistical models. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 217–232.Google Scholar
  46. Dawson, D. A. and Hochberg, K. J. (1982). Wandering random measures in the Fleming-Viot model. Ann. Probability 10, 554–580.MathSciNetzbMATHCrossRefGoogle Scholar
  47. Dellacherie, C. and Meyer, P.-A. (1975, 1980). Probabilités et Potentiel, Ch. I–IV; Ch. V–VIII. Herman, Paris.zbMATHGoogle Scholar
  48. Diaconis, P. (1977). Finite forms of de Finetti's theorem on exchangeability. Synthese 36, 271–281.MathSciNetzbMATHCrossRefGoogle Scholar
  49. Diaconis, P. and Freedman, D. (1980a). Finite exchangeable sequences. Ann. Probability 8, 745–764.MathSciNetzbMATHCrossRefGoogle Scholar
  50. Diaconis, P. and Freedman, D. (1980b). de Finetti's theorem for Markov chains. Ann. Probability 8, 115–130.MathSciNetzbMATHCrossRefGoogle Scholar
  51. Diaconis, P. and Freedman, D. (1980c). de Finetti's generalizations of exchangeability. In Studies in Inductive Logic and Probability II (R. C. Jeffrey, ed.). University of California Press, Berkeley.Google Scholar
  52. Diaconis, P. and Freedman, D. (1981). On the statistics of vision: the Julesz conjecture. J. Math. Psychol. 24, 112–138.MathSciNetzbMATHCrossRefGoogle Scholar
  53. Diaconis, P. and Freedman, D. (1982). Partial exchangeability and sufficiency. Preprint, Department of Statistics, Stanford University (to appear in Sankhyā).Google Scholar
  54. Diaconis, P. and Ylvisaker, Y. (1979). Conjugate priors for exponential families. Ann. Statist. 7, 269–281.MathSciNetzbMATHCrossRefGoogle Scholar
  55. Dohler, R. (1980). On the conditional independence of random events. Theory Probability Appl. 25, 628–634.MathSciNetzbMATHCrossRefGoogle Scholar
  56. Doob, J. L. (1953). Stochastic Processes. Wiley, New York.zbMATHGoogle Scholar
  57. Dubins, L. E. (1982). Towards characterizing the set of ergodic probabilities. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 61–74.Google Scholar
  58. Dubins, L. E. (1983). Some exchangeable probabilities which are singular with respect to all presentable probabilities. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete 64, 1–6.MathSciNetzbMATHCrossRefGoogle Scholar
  59. Dubins, L. E. and Freedman, D. (1979). Exchangeable processes need not be mixtures of independent identically distributed random variables. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete 48, 115–132.MathSciNetzbMATHCrossRefGoogle Scholar
  60. Dynkin, E. B. (1978). Sufficient statistics and extreme points. Ann. Probability 6, 705–730.MathSciNetzbMATHCrossRefGoogle Scholar
  61. Eagleson, G. K. (1979). A Poisson limit theorem for weakly exchangeable events, J. Appl. Prob. 16, 794–802.MathSciNetzbMATHCrossRefGoogle Scholar
  62. Eagleson, G. K. (1982). Weak limit theorems for exchangeable random variables. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 251–268.Google Scholar
  63. Eagleson, G. K. and Weber, N. C. (1978). Limit theorems for weakly exchangeable arrays. Math. Proc. Cambridge Phil. Soc. 84, 123–130.MathSciNetzbMATHCrossRefGoogle Scholar
  64. Eaton, M. (1981). On the projections of isotropic distributions. Ann. Statist. 9, 391–400.MathSciNetzbMATHCrossRefGoogle Scholar
  65. Feller, W. (1971). An Introduction to Probability Theory and its Applications, vol. 2, Wiley, New York.zbMATHGoogle Scholar
  66. Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1, 209–230.MathSciNetzbMATHCrossRefGoogle Scholar
  67. Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2, 615–629.MathSciNetzbMATHCrossRefGoogle Scholar
  68. Figiel, T. and Sucheston, L. (1976). An application of Ramsey sets in analysis. Advances in Math. 20, 103–105.MathSciNetzbMATHCrossRefGoogle Scholar
  69. de Finetti, B. (1937). La prevision: ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincaré 7, 1–68.MathSciNetzbMATHGoogle Scholar
  70. de Finetti, B. (1938). Sur la condition d'equivalence partielle. Trans. in Studies in Inductive Logic and Probability II (R. C. Jeffrey, ed.). University of California Press, Berkeley.Google Scholar
  71. de Finetti, B. (1972). Probability, Induction and Statistics. Wiley, New York.zbMATHGoogle Scholar
  72. de Finetti, B. (1974). Theory of Probability. Wiley, New York.zbMATHGoogle Scholar
  73. Freedman, D. (1962a). Mixtures of Markov processes. Ann. Math. Statist. 33, 114–118.MathSciNetzbMATHCrossRefGoogle Scholar
  74. Freedman, D. (1962b). Invariants under mixing which generalize de Finetti's theorem. Ann. Math. Statist. 33, 916–923.MathSciNetzbMATHCrossRefGoogle Scholar
  75. Freedman, D. (1963). Invariants under mixing which generalize de Finetti's theorem: continuous time parameter. Ann. Math. Statist. 34, 1194–1216.MathSciNetzbMATHCrossRefGoogle Scholar
  76. Freedman, D. (1980). A mixture of independent identically distributed random variables need not admit a regular conditional probability given the exchangeable σ-field. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete 51, 239–248.MathSciNetzbMATHCrossRefGoogle Scholar
  77. Freedman, D. and Diaconis, P. (1982). de Finetti's theorem for symmetric location families. Ann. Statist. 10, 184–189.MathSciNetzbMATHCrossRefGoogle Scholar
  78. Galambos, J. (1982). The role of exchangeability in the theory of order statistics. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 75–86.Google Scholar
  79. Grigorenko, L. A. (1979). On the σ-algebra of symmetric events for a countable Markov chain. Theor. Probability Appl. 24, 199–204.MathSciNetzbMATHCrossRefGoogle Scholar
  80. Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and its Application, Academic Press, New York.zbMATHGoogle Scholar
  81. Heath, D. and Sudderth, W. (1976). de Finetti's theorem for exchangeable random variables. Amer. Statistician 30, 188–189.MathSciNetzbMATHGoogle Scholar
  82. Helland, I. S. (1982). Central limit theorems for martingales with discrete or continuous time. Scand. J. Statist. 9, 79–94.MathSciNetzbMATHGoogle Scholar
  83. Heller, A. (1965). On stochastic processes derived from Markov chains. Ann. Math. Statist. 36, 1286–1291.MathSciNetzbMATHCrossRefGoogle Scholar
  84. Hewitt, E. and Savage, L. J. (1955) Symmetric measures on Cartesian products. Trans. Amer. Math. Soc. 80, 470–501.MathSciNetzbMATHCrossRefGoogle Scholar
  85. Hida, T. (1980). Brownian Motion. Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  86. Hoover, D. N. (1979). Relations on probability spaces and arrays of random variables. Preprint.Google Scholar
  87. Hoover, D. N. (1982). Row-column exchangeability and a generalized model for exchangeability. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 281–291.Google Scholar
  88. Horowitz, J. (1972). Semilinear Markov processes, subordinators and renewal theory. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete 24, 167–193.MathSciNetzbMATHCrossRefGoogle Scholar
  89. Hu, Y.-S. (1979). A note on exchangeable events. J. Appl. Prob. 16, 662–664.MathSciNetzbMATHCrossRefGoogle Scholar
  90. Ignatov, Ts. (1982). On a constant arising in the theory of symmetric groups and on Poisson-Dirichlet measures. Theory Prob. Appl. 27, 136–147.MathSciNetzbMATHCrossRefGoogle Scholar
  91. Isaac, R. (1977). Generalized Hewitt-Savage theorems for strictly stationary processes. Proc. Amer. Math. Soc. 63, 313–316.MathSciNetzbMATHCrossRefGoogle Scholar
  92. Jaynes, E. (1983). Some applications and extensions of the de Finetti representation theorem. In Bayesian Inference and Decision Techniques (P. K. Goel and A. Zellnor, eds.). North-Holland, Amsterdam.Google Scholar
  93. Johnson, N. L. and Kotz, S. (1977). Urn Models and their Applications. Wiley, New York.zbMATHGoogle Scholar
  94. Kallenberg, O. (1973). Canonical representations and convergence criteria for processes with interchangeable increments. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete 27, 23–36.MathSciNetzbMATHCrossRefGoogle Scholar
  95. Kallenberg, O. (1974). Path properties of processes with independent and interchangeable increments. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete 28, 257–271.MathSciNetzbMATHCrossRefGoogle Scholar
  96. Kallenberg, O. (1975). Infinitely divisible processes with interchangeable increments and random measures under convolution. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete 32, 309–321.MathSciNetzbMATHCrossRefGoogle Scholar
  97. Kallenberg, O. (1976). Random Measures. Akademie-Verlag, Berlin.zbMATHGoogle Scholar
  98. Kallenberg, O. (1976–81). On the structure of stationary flat processes, I–III. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete 37, 157–174; 52, 127–147; 56, 239–253.MathSciNetzbMATHCrossRefGoogle Scholar
  99. Kallenberg, O. (1982a). Characterizations and embedding properties in exchangeability. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete 60, 249–281.MathSciNetzbMATHCrossRefGoogle Scholar
  100. Kallenberg, O. (1982b). A dynamical approach to exchangeability. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 87–96.Google Scholar
  101. Kallenberg, O. (1982c). The stationary-invariance problem. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 293–295.Google Scholar
  102. Kallenberg, O. (1982d). Conditioning in point processes. Preprint, University of Göteborg.Google Scholar
  103. Kallenberg, O. (1983). The local time intensity of an exchangeable interval partition. Preprint, University of Göteborg.Google Scholar
  104. Kelly, F. P. (1979). Reversibility and Stochastic Networks. Wiley, New York.zbMATHGoogle Scholar
  105. Kemperman, J. H. B. (1973). Moment problems for sampling without replacement I. Indagationes Math. 35, 149–164.MathSciNetzbMATHCrossRefGoogle Scholar
  106. Kendall, D. G. (1967). On finite and infinite sequences of exchangeable events. Studia Scient. Math. Hung. 2, 319–327.MathSciNetzbMATHGoogle Scholar
  107. Kerov, S. V. and Vershik, A. M. (1982). Characters of infinite symmetric groups and probability properties of Robinson-Shenstead-Knuth's algorithm. Preprint.Google Scholar
  108. Kimberling, C. H. (1973). Exchangeable events and completely monotonic sequences. Rocky Mtn. J. 3, 565–574.MathSciNetzbMATHCrossRefGoogle Scholar
  109. Kindermann, R. and Snell, J. L. (1980). Markov Random Fields and their Applications. American Math. Soc., Providence.zbMATHCrossRefGoogle Scholar
  110. Kingman, J. F. C. (1978a). Uses of exchangeability. Ann. Probability 6, 183–197.MathSciNetzbMATHCrossRefGoogle Scholar
  111. Kingman, J. F. C. (1978b). The representation of partition structures. J. Lond. Math. Soc. 18, 374–380.MathSciNetzbMATHCrossRefGoogle Scholar
  112. Kingman, J. F. C. (1978c). Random partitions in population genetics. Proc. Roy. Soc. 361, 1–20.MathSciNetzbMATHCrossRefGoogle Scholar
  113. Kingman, J. F. C. (1980). Mathematics of Genetic Diversity. SIAM, Philadelphia.zbMATHCrossRefGoogle Scholar
  114. Kingman, J. F. C. (1982a). Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 97–112.Google Scholar
  115. Kingman, J. F. C. (1982b). The coalescent. Stochastic Processes Appl. 13, 235–248.MathSciNetzbMATHCrossRefGoogle Scholar
  116. Knuth, D. E. (1981). The Art of Computer Programming, Vol. 2, 2nd Ed. Addison-Wesley, Reading, Mass.zbMATHGoogle Scholar
  117. Kolchin, V. F. and Sevast'yanov, B. A. (1978). Random Allocations. Winston, New York.zbMATHGoogle Scholar
  118. Komlós, J. (1967). A generalisation of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18, 217–229.MathSciNetzbMATHCrossRefGoogle Scholar
  119. Lauritzen, S. L. (1982). Statistical Models as Extremal Families. Aalborg University Press.Google Scholar
  120. Letac, G. (1981a). Isotropy and sphericity: some characterizations of the normal distribution. Ann. Statist. 9, 408–417.MathSciNetzbMATHCrossRefGoogle Scholar
  121. Letac, G. (1981b). Problèmes classiques de Probabilité sur un couple de Gelfand. In Analytic Methods in Probability Theory (D. Dugué et al., eds.). Springer Lecture Notes in Mathematics 861, New York.Google Scholar
  122. Loève, M. (1960). Probability Theory. Van Nostrand, Princeton.zbMATHGoogle Scholar
  123. Lynch, J. (1982a). On a representation for row-column-exchangeable arrays. Technical Report 82, Department of Mathematics, University of South Carolina.Google Scholar
  124. Lynch, J. (1982b). Canonical row-column-exchangeable arrays. Technical Report 84, Department of Mathematics, University of South Carolina.Google Scholar
  125. Maitra, A. (1977). Integral representations of invariant measures. Trans. Amer. Math. Soc. 229, 209–225.MathSciNetzbMATHCrossRefGoogle Scholar
  126. Mandelbaum, A. and Taqqu, M. S. (1983). Invariance principle for symmetric statistics. Preprint, Operations Research, Cornell University.Google Scholar
  127. Mansour, B. (1981). Le cube comme couple de Gelfand. Thesis, University of Toulouse.Google Scholar
  128. Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and its Application. Academic Press, New York.zbMATHGoogle Scholar
  129. Milgrom, P. R. and Weber, R. J. (1983). Exchangeable affiliated random variables. Preprint.Google Scholar
  130. Moran, P. A. P. (1973). A central limit theorem for exchangeable variates with geometrical applications. J. Appl. Prob. 10, 837–846.zbMATHCrossRefGoogle Scholar
  131. O'Brien, G. L. and Vervaat, W. (1983). Self-similar processes with stationary increments generated by point processes. Preprint.Google Scholar
  132. Olshen, R. A. (1971). The coincidence of measure algebras under an exchangeable probability. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete 18, 153–158.MathSciNetzbMATHCrossRefGoogle Scholar
  133. Olshen, R. A. (1973). A note on exchangeable sequences. Zeitschrift fur Wahrscheinlichkeitstheorie verw. Gebiete 28, 317–321.MathSciNetzbMATHCrossRefGoogle Scholar
  134. Ornstein, D. S. (1973). An application of ergodic theory to probability theory. Ann. Probability 1, 43–65.MathSciNetzbMATHCrossRefGoogle Scholar
  135. Palacios, J. (1982). The exchangeable σ-field of Markov chains. Ph. D. thesis, University of California, Berkeley.Google Scholar
  136. Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Academic Press, New York.zbMATHCrossRefGoogle Scholar
  137. Pavlov, Y. L. (1981). Limit theorem for a characteristic of a random mapping. Theory Prob. Appl. 26, 829–834.zbMATHCrossRefGoogle Scholar
  138. Pfeiffer, P. E. (1979). Conditional Independence in Applied Probability. UMAP, Educational Development Center, Newton, Mass.zbMATHGoogle Scholar
  139. Phelps, R. R. (1966). Lectures on Choquet's Theorem. Van Nostrand, Princeton.zbMATHGoogle Scholar
  140. Pitman, J. (1978). An extension of de Finetti's theorem. Adv. Appl. Prob. 10, 268–270.CrossRefGoogle Scholar
  141. Pittel, B. (1983). On the distributions related to transitive classes of random finite mappings. Ann. Probability 11, 428–441.MathSciNetzbMATHCrossRefGoogle Scholar
  142. Quine, M. P. (1978). A functional central limit theorem for a class of exchangeable sequences. Preprint, University of Sydney.Google Scholar
  143. Quine, M. P. (1979). A functional central limit theorem for a generalized occupancy problem. Stochastic Proc. Appl. 9, 109–115.MathSciNetzbMATHCrossRefGoogle Scholar
  144. Rényi, A. (1963). On stable sequences of events. Sankhyā Ser. A 25, 293–302.MathSciNetzbMATHGoogle Scholar
  145. Rényi, A. and Révész, P. (1963). A study of sequences of equivalent events as special stable sequences. Publ. Math. Debrecen. 10, 319–325.MathSciNetzbMATHGoogle Scholar
  146. Ressel, P. (1983). de Finetti type thorems: an analytical approach. Preprint, Department of Statistics, Stanford University.Google Scholar
  147. Ridler-Rowe, C. J. (1967). On two problems on exchangeable events. Studia Sci. Math. Hungar. 2, 415–418.MathSciNetzbMATHGoogle Scholar
  148. Ryll-Nardzewski, C. (1957). On stationary sequences of random variables and the de Finetti's equivalence. Colloquium Math. 4, 149–156.MathSciNetzbMATHGoogle Scholar
  149. Saunders, R. (1976). On joint exchangeability and conservative processes with stochastic rates. J. Appl. Prob. 13, 584–590.MathSciNetzbMATHCrossRefGoogle Scholar
  150. Schoenberg, I. J. (1938). Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44, 522–536.MathSciNetzbMATHCrossRefGoogle Scholar
  151. Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.zbMATHCrossRefGoogle Scholar
  152. Shaked, M. and Tong, Y. L. (1983). Some partial orderings of exchangeable random variables by positive dependence. Preprint, Mathematics Department, University of Arizona.Google Scholar
  153. Shields, P. C. (1979). Stationary coding of processes. IEEE Trans. Information Theory 25, 283–291.MathSciNetzbMATHCrossRefGoogle Scholar
  154. Sigalotti, L. (1982). On particular renewal processes interesting reliability theory. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 303–311.Google Scholar
  155. Slud, E. V. (1978). A note on exchangeable sequences of events. Rocky Mtn. J. 8, 439–442.MathSciNetzbMATHCrossRefGoogle Scholar
  156. Smith, A. M. F. (1981). On random sequences with centered spherical symmetry. J. Roy. Statist. Soc. Ser. B 43, 208–209.MathSciNetzbMATHGoogle Scholar
  157. Speed, T. P. (1982). Cumulants, k-statistics and their generalisations. Preprint, University of Western Australia.Google Scholar
  158. Spitzer, F. (1975). Markov random fields on an infinite tree. Ann. Probability 3, 387–398.MathSciNetzbMATHCrossRefGoogle Scholar
  159. Spizzichino, F. (1982). Extendibility of symmetric probability distributions and related bounds. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 313–320.Google Scholar
  160. Stein, C. (1983). Approximate computation of expectations. Unpublished lecture notes.Google Scholar
  161. Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439.MathSciNetzbMATHCrossRefGoogle Scholar
  162. Surabian, C. G. (1979). A survey of exchangeability. M.Sc. thesis, Tufts University.Google Scholar
  163. Taqqu, M. S. (1982). Self-similar processes and related ultraviolet and infrared catastrophes. In Random Fields (J. Fritz, ed.). North-Holland, Amsterdam.Google Scholar
  164. Teicher, H. (1982). Renewal theory for interchangeable random variables. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.). North-Holland, Amsterdam, 113–121.Google Scholar
  165. Troutman, B. M. (1983). Weak convergence of the adjusted range of cumulative sums of exchangeable random variables. J. Appl. Prob. 20, 297–304.MathSciNetzbMATHCrossRefGoogle Scholar
  166. Vershik, A. M. and Schmidt, A. A. (1977). Limite measures arising in the theory of symmetric groups I. Theory Prob. Appl. 22, 70–85.CrossRefGoogle Scholar
  167. Watterson, G. A. (1976). The stationary distribution of the infinite-many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651.MathSciNetzbMATHCrossRefGoogle Scholar
  168. Weber, N. C. (1980). A martingale approach to central limit theorems for exchangeable random variables. J. Appl. Prob. 17, 662–673.MathSciNetzbMATHCrossRefGoogle Scholar
  169. Zabell, S. L. (1982). W. E. Johnson's "sufficientness" postulate. Ann. Statist. 10, 1091–1099.MathSciNetzbMATHCrossRefGoogle Scholar
  170. Zachary, S. (1983). Countable state space Markov random fields and Markov chains on trees. Ann. Probability. to appear.Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • David J. Aldous

There are no affiliations available

Personalised recommendations