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Processes with Block-Associated Increments

Part of: Stochastic processes Distribution theory - Probability Limit theorems

Published online by Cambridge University Press:  14 July 2016

Adam Jakubowski  and
Joanna Karłowska-Pik
Adam Jakubowski*
Affiliation:
Nicolaus Copernicus University
Joanna Karłowska-Pik*
Affiliation:
Nicolaus Copernicus University
*
∗Postal address: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland.
∗Postal address: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland.
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Abstract

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This paper is motivated by relations between association and independence of random variables. It is well known that, for real random variables, independence implies association in the sense of Esary, Proschan and Walkup (1967), while, for random vectors, this simple relationship breaks. We modify the notion of association in such a way that any vector-valued process with independent increments also has associated increments in the new sense - association between blocks. The new notion is quite natural and admits nice characterization for some classes of processes. In particular, using the covariance interpolation formula due to Houdré, Pérez-Abreu and Surgailis (1998), we show that within the class of multidimensional Gaussian processes, block association of increments is equivalent to supermodularity (in time) of the covariance functions. We also define corresponding versions of weak association, positive association, and negative association. It turns out that the central limit theorem for weakly associated random vectors due to Burton, Dabrowski and Dehling (1986) remains valid, if the weak association is relaxed to the weak association between blocks.

Keywords

Associationindependenceassociated incrementsindependent incrementsweak associationsupermodularity

MSC classification

Secondary: 60G99: None of the above, but in this section 60G15: Gaussian processes 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 514 - 526
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Barlow, R. E. and Proschan, F. (1996). Mathematical Theory of Reliability. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
[2] Bulinski, A. and Shashkin, A. (2007). Limit Theorems for Associated Random Fields and Related Systems. World Scientific, Hackensack, NJ.Google Scholar
[3] Burton, R. M., Dabrowski, A. R. and Dehling, H. (1986). An invariance principle for weakly associated random vectors. Stoch. Process. Appl. 23, 301–306.CrossRefGoogle Scholar
[4] Dabrowski, A. R. and Jakubowski, A. (1994). Stable limits for associated random variables. Ann. Prob. 22, 1–16.Google Scholar
[5] Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.Google Scholar
[6] Glasserman, P. (1992). Processes with associated increments. J. Appl. Prob. 29, 313–333.Google Scholar
[7] Houdré, C., Pérez-Abreu, V. and Surgailis, D. (1998). Interpolation, correlation identities, and inequalities for infinitely divisible variables. J. Fourier Anal. Appl. 4, 651–668.Google Scholar
[8] Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables, with applications. Ann. Statist. 11, 286–295.CrossRefGoogle Scholar
[9] Karłowska-Pik, J. and Schreiber, T. (2008). Association criteria for M-infinitely-divisible and U-infinitely-divisible random sets. Prob. Math. Statist. 28, 169–178.Google Scholar
[10] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.Google Scholar
[11] Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
[12] Maruyama, G. (1970). Infinitely divisible processes. Teor. Verojat. Primen. 15, 3–23 (in Russian). English translation: Theory Prob. Appl. 15, 1-22.Google Scholar
[13] Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
[14] Newman, C. M. (1980). Normal fluctuations and the FKG inequalities. Commun. Math. Phys. 74, 119–128.CrossRefGoogle Scholar
[15] Newman, C. M. (1983). A general central limit theorem for FKG systems. Commun. Math. Phys. 91, 75–80.Google Scholar
[16] Newman, C.M., and Wright, A. L. (1981). An invariance principle for certain dependent sequences. Ann. Prob. 9, 671–675.Google Scholar
[17] Pitt, L. D. (1982). Positively correlated normal variables are associated. Ann. Prob. 10, 496–499.Google Scholar
[18] Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451–487.Google Scholar
[19] Resnick, S. I. (1988). Association and multivariate extreme value distributions. In: Studies in Statistical Modelling and Statistical Science, ed. Heyde, C. C., Statistical Society of Australia, pp. 261–271.Google Scholar
[20] Samorodnitsky, G. (1995). Association of infintely divisible random vectors. Stoch. Process. Appl. 55, 45–55.Google Scholar
[21] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge.Google Scholar