Abstract
We consider random arrays and the associated empirical distributions obtained by multivariate sampling from a stationary process. Under suitable conditions, one gets convergence toward a separately exchangeable array and its ergodic distribution. The result is related to the statistical problem of estimating the representing function of an exchangeable array. The latter problem is well-posed only for shell-measurable arrays, where the grid processes based on finite sub-arrays form consistent estimates with respect to a suitable norm. In general, the required consistency holds only in the distributional sense for the generated arrays.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Aldous, D. J. (1981). Representations for partially exchangeable arrays of random variables. J. Multivar. Anal. 11, 581–598.
Aldous, D. J. (1985). Exchangeability and related topics. École d'Été de Prob. Saint-Flour XIII, Springer, Berlin. LNM 1117, 1–198.
Dellacherie, C., and Meyer, P. A. (1975). Probabilités et Potentiel, Chap. I–IV. Hermann, Paris.
Diaconis, P., and Freedman, D. (1981). On the statistics of vision: The Julesz conjecture. J. Math. Psychol. 24, 112–138.
Hoover, D. N. (1979). Relations on probability spaces and arrays of random variables. Institute of Advanced Study, Princeton. Preprint.
Hoover, D. N. (1982). Row-column exchangeability and a generalized model for probability. In Koch, G., and Spizzichino, F. (eds.), Exchangeability in Probability and Statistics, North-Holland, Amsterdam, pp. 281–291.
Hoover, D. N. (1989). Tail fields of partially exchangeable arrays. J. Multivar. Anal. 31, 160–163.
Kallenberg, O. (1989). On the representation theorem for exchangeable arrays. J. Multivar. Anal. 30, 137–154.
Kallenberg, O. (1992). Symmetries on random arrays and set-indexed processes. J. Theor. Prob. 5, 727–765.
Kallenberg, O. (1995). Random arrays and functionals with multivariate rotational symmetries. Prob. Th. Rel. Fields 103, 91–141.
Kallenberg, O. (1997). Foundations of Modern Probability, Springer, New York.
Kallenberg, O. (1999). Asymptotically invariant sampling and averaging from stationary-like processes. Stoch. Proc. Appl. 82 (to appear).
Krengel, U. (1985). Ergodic Theorems, de Gruyter, Berlin.
Matthes, K., Kerstan, J., and Mecke, J. (1978). Infinitely Divisible Point Processes, Wiley, Chichester.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kallenberg, O. Multivariate Sampling and the Estimation Problem for Exchangeable Arrays. Journal of Theoretical Probability 12, 859–883 (1999). https://doi.org/10.1023/A:1021692202530
Issue Date:
DOI: https://doi.org/10.1023/A:1021692202530