## Journal of Applied Probability

### Processes with block-associated increments

#### Abstract

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.

#### Article information

Source
J. Appl. Probab., Volume 48, Number 2 (2011), 514-526.

Dates
First available in Project Euclid: 21 June 2011

https://projecteuclid.org/euclid.jap/1308662641

Digital Object Identifier
doi:10.1239/jap/1308662641

Mathematical Reviews number (MathSciNet)
MR2840313

Zentralblatt MATH identifier
1235.60056

#### Citation

Jakubowski, Adam; Karłowska-Pik, Joanna. Processes with block-associated increments. J. Appl. Probab. 48 (2011), no. 2, 514--526. doi:10.1239/jap/1308662641. https://projecteuclid.org/euclid.jap/1308662641

#### References

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