Journal of Applied Probability

Processes with block-associated increments

Adam Jakubowski and Joanna Karłowska-Pik

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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

Permanent link to this document
https://projecteuclid.org/euclid.jap/1308662641

Digital Object Identifier
doi:10.1239/jap/1308662641

Mathematical Reviews number (MathSciNet)
MR2840313

Zentralblatt MATH identifier
1235.60056

Subjects
Primary: 60G99: None of the above, but in this section
Secondary: 60G15: Gaussian processes 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems

Keywords
Association independence associated increments independent increments weak association supermodularity

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


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