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Positive Dependence and Weak Convergence

Part of: Probability theory on algebraic and topological structures Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

A. Colangelo ,
A. Müller  and
M. Scarsini
A. Colangelo*
Affiliation:
Università dell'Insubria
A. Müller*
Affiliation:
Universität Karlsruhe
M. Scarsini*
Affiliation:
Università di Torino
*
∗ Postal address: Dipartimento di Economia, Università dell'Insubria, Via Monte Generoso 71, I-21100 Varese, Italy. Email address: antonio.colangelo@uninsubria.it
∗∗ Postal address: Institut für Wirtschaftstheorie und Operations Research, Universität Karlsruhe, Geb. 20.21, D-76128 Karlsruhe, Germany. Email address: mueller@wior.uni-karlsruhe.de
∗∗∗ Postal address: Dipartimento di Statistica e Matematica Applicata, Università di Torino, Piazza Arbarello 8, I-10122 Torino, Italy. Email address: marco.scarsini@unito.it
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Abstract

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A more general definition of MTP2 (multivariate total positivity of order 2) probability measure is given, without assuming the existence of a density. Under this definition the class of MTP2 measures is proved to be closed under weak convergence. Characterizations of the MTP2 property are proved under this more general definition. Then a precise definition of conditionally increasing measure is provided, and closure under weak convergence of the class of conditionally increasing measures is proved. As an application we investigate MTP2 properties of stationary distributions of Markov chains, which are of interest in actuarial science.

Keywords

AffiliationMTP2conditionally increasing in sequenceconditionally increasing

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60B10: Convergence of probability measures

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 1 , March 2006 , pp. 48 - 59
Copyright
© Applied Probability Trust 2006 

References

Alam, K. and Wallenius, K. T. (1976). Positive dependence and monotonicity in conditional distributions. Commun. Statist. Theory Meth. 5, 525–534.Google Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
Block, H. W., Savits, T. H. and Shaked, M. (1982). Some concepts of negative dependence. Ann. Prob. 10, 765–772.CrossRefGoogle Scholar
Borgan, Ø., Hoem, J. M. and Norberg, R. (1981). A nonasymptotic criterion for the evaluation of automobile bonus systems. Scand. Actuarial J. 1981, 165–178.Google Scholar
Colangelo, A., Scarsini, M. and Shaked, M. (2005). Some notions of multivariate positive dependence. Insurance Math. Econom. 37, 13–26.Google Scholar
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.Google Scholar
Kamae, T., Krengel, U. and O'Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal. 10, 467–498.Google Scholar
Kijima, M. (1997). Markov Processes for Stochastic Modeling. Chapman and Hall, London.Google Scholar
Kimeldorf, G. and Sampson, A. R. (1987). Positive dependence orderings. Ann. Inst. Statist. Math. 39, 113–128.CrossRefGoogle Scholar
Kimeldorf, G. and Sampson, A. R. (1989). A framework for positive dependence. Ann. Inst. Statist. Math. 41, 31–45.Google Scholar
Lehmann, E. L. (1966). Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153.Google Scholar
Milgrom, P. R. and Weber, R. J. (1982). A theory of auctions and competitive bidding. Econometrica 50, 1089–1122.Google Scholar
Müller, A. (1997). Stochastic orders generated by integrals: a unified study. Adv. Appl. Prob. 29, 414–428.Google Scholar
Müller, A. and Scarsini, M. (2001). Stochastic comparison of random vectors with a common copula. Math. Operat. Res. 26, 723–740.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Norberg, R. (1976). A credibility theory for automobile bonus systems. Scand. Actuarial J. 1976, 92–107.Google Scholar
Pellerey, F. and Semeraro, P. (2003). A positive dependence notion based on the supermodular order. Res. Rep. 6, Dipartimento di Matematica, Politecnico di Torino.Google Scholar
Sarkar, S. K. (1998). Some probability inequalities for ordered MTP2 random variables: a proof of the Simes conjecture. Ann. Statist. 26, 494–504.Google Scholar
Shiryaev, A. N. (1996). Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Tukey, J. W. (1958). A problem of Berkson, and minimum variance orderly estimators. Ann. Math. Statist. 29, 588–592.Google Scholar