Random assignment processes: strong law of large numbers and De Finetti theorem Original Paper
First Online: 06 September 2014 Received: 28 November 2013 Accepted: 27 July 2014 DOI :
10.1007/s11749-014-0396-0
Cite this article as: Vélez, R. & Prieto-Rumeau, T. TEST (2015) 24: 136. doi:10.1007/s11749-014-0396-0 Abstract In the framework of a random assignment process—which randomly assigns an index within a finite set of labels to the points of an arbitrary set—we study sufficient conditions for a strong law of large numbers and a De Finetti theorem. In particular, this yields a family of finite-valued nonexchangeable random variables that are conditionally independent given some other random variable, that is, they verify a De Finetti theorem. We show an application of the De Finetti theorem and the law of large numbers to an estimation problem.
Keywords Random assignment processes Exchangeability Strong laws of large numbers De Finetti theorem This research was supported by the Spanish Ministerio de Economía y Competitividad, Grant Number MTM2012-31393.
References Aldous DJ (1981) Representations for partially exchangeable arrays of random variables. J Multivariate Anal 11:581–598
CrossRef MATH MathSciNet Google Scholar Aldous DJ (1982) On exchangeability and conditional independence. In: Exchangeability in probability and statistics, Rome, 1981. North-Holland, Amsterdam-New York, pp 165–170
Aldous DJ (1985) Exchangeability and related topics. Lecture Notes in Math 1117. Springer, Berlin
Google Scholar Chow YS, Teicher H (1988) Probability theory. Independence, interchangeability, martingales. Springer-Verlag, New York
MATH Google Scholar De Finetti B (1930) Funzione caratteristica di un fenomeno aleatorio. Memorie della R Accademia dei Lincei IV:86–133
Google Scholar De Finetti B (1937) La prévision: ses lois logiques, ses sources subjectives. Ann Inst H Poincaré 7:1–68
Google Scholar Diaconis P, Freedman D (1980) Finite exchangeable sequences. Ann Probab 8:745–764
CrossRef MATH MathSciNet Google Scholar Diaconis P, Freedman D (1980) De Finetti’s theorem for Markov chains. Ann Probab 8:115–130
CrossRef MATH MathSciNet Google Scholar Diaconis P, Freedman D (1984) Partial exchangeability and sufficiency. In: Statistics: applications and new directions, Calcutta, 1981. Indian Statist Inst, Calcutta, India, pp 205–236
Dovbysh LN, Sudakov VN (1982) Gram-de Finetti matrices. J Soviet Math 24:3047–3054
Google Scholar Dubins LE, Freedman DA (1979) Exchangeable processes need not be mixtures of independent, identically distributed random variables. Z Wahrsch Verw Gebiete 48:115–132
CrossRef MATH MathSciNet Google Scholar Dynkin EB (1978) Sufficient statistics and extreme points. Ann Probab 6:705–730
CrossRef MATH MathSciNet Google Scholar Feller W (1966) An introduction to probability theory and its applications, vol II, 2nd edn. John Wiley, New York
Freedman DA (1996) De Finetti’s theorem in continuous time. In: Statistics, probability and game theory, IMS Lecture Notes Monogr Ser 30. Inst Math Statist, Hayward, CA, pp 83–98
Fristedt B, Gray L (1997) A modern approach to probability theory. Birkhäuser, Boston
CrossRef MATH Google Scholar Gnedin AV (1997) The representation of composition structures. Ann Probab 25:1437–1450
CrossRef MATH MathSciNet Google Scholar Gnedin AV, Pitman J (2004) Regenerative partition structures. Electron J Combin 11(2):R12
MathSciNet Google Scholar Hewitt E, Savage LJ (1955) Symmetric measures on cartesian products. Trans Am Math Soc 80:470–501
CrossRef MATH MathSciNet Google Scholar Hoover DN (1982) Row–column exchangeability and a generalized model for probability. In: Exchangeability in probability and statistics, Rome, 1981. North-Holland, Amsterdam-New York, pp 281–291
Kallenberg O (1973) A canonical representation of symmetrically distributed random measures. In: Mathematics and statistics (in honour of Harald Bergström on the occasion of his 65th birthday). Department of Math, Chalmers Inst Tech, Göteborg, pp 41–48
Kallenberg O (1975) On symmetrically distributed random measures. Trans Am Math Soc 202:105–121
CrossRef MATH MathSciNet Google Scholar Kallenberg O (1976) Random measures. Academic Press, London-New York
MATH Google Scholar Kallenberg O (1982) Characterizations and embedding properties in exchangeability. Z Wahrsch Verw Gebiete 60:249–281
CrossRef MATH MathSciNet Google Scholar Kallenberg O (1982) A dynamical approach to exchangeability. In: Exchangeability in probability and statistics, Rome, 1981. North-Holland, Amsterdam-New York, pp 87–96
Kallenberg O (1989) On the representation theorem for exchangeable arrays. J Multivariate Anal 30: 137–154
Kallenberg O (1992) Symmetries on random arrays and set-indexed processes. J Theoret Probab 5:727–765
CrossRef MATH MathSciNet Google Scholar Kallenberg O (1997) Foundations of modern probability. Springer-Verlag, New York
MATH Google Scholar Kallenberg O (2005) Probabilistic symmetries and invariance principles. Springer, New York
MATH Google Scholar Kerns GJ, Székely GJ (2006) De Finetti’s theorem for abstract finite exchangeable sequences. J Theoret Probab 19:589–608
CrossRef MATH MathSciNet Google Scholar Kingman JFC (1978) The representation of partition structures. J London Math Soc 18:374–380
CrossRef MATH MathSciNet Google Scholar Lauritzen SL (1982) Statistical models as extremal families. Aalborg University Press, Aalborg
MATH Google Scholar Loève M (1977) Probability theory I, 4th edn. Springer, New York
MATH Google Scholar Panchenko D (2010) On the Dovbysh–Sudakov representation result. Electron Commun Probab 15:330–338
CrossRef MATH MathSciNet Google Scholar Pitman JW (1978) An extension of de Finetti’s Theorem. Adv Appl Probab 10:268–270
CrossRef Google Scholar Pitman J (1995) Exchangeable and partially exchangeable random partitions. Probab Theory Related Fields 102:145–158
CrossRef MATH MathSciNet Google Scholar Vélez R, Prieto-Rumeau T (2008) A De Finetti-type theorem for nonexchangeable finite-valued random variables. J Math Anal Appl 347:407–415
CrossRef MATH MathSciNet Google Scholar Vélez R, Prieto-Rumeau T (2009) De Finetti’s-type results for some families of non identically distributed random variables. Electron J Probab 14:72–86
MATH MathSciNet Google Scholar Vélez R, Prieto-Rumeau T (2010) De Finetti-type theorems for random selection processes. Necessary and sufficient conditions. J Math Anal Appl 365:198–209
CrossRef MATH MathSciNet Google Scholar Vélez R, Prieto-Rumeau T (2011) De Finetti’s type theorems for nonexchangeable 0–1 random variables. Test 20:293–310
CrossRef MATH MathSciNet Google Scholar © Sociedad de Estadística e Investigación Operativa 2014
Authors and Affiliations 1. Statistics Deparment UNED Madrid Spain