Summary
Consider an array X = (X ij ,i,j∈N) of random variables, and let U=(U ij ) and V=(V ij ) be orthogonal transformations affecting only finitely many coordinates. Say that X is separately rotatable if \(UXV^T \mathop = \limits^d X\) for arbitrary U and V, and jointly rotatable if this holds with U=V. Restricting U and V to the class of permutations, we get instead the property of separate or joint exchangeability. Processes on ℝ 2+ , ℝ+ × [0,1] or [0, 1]2 are said to be separately or jointly exchangeable, if the arrays of increments over arbitrary square grids have these properties. For some of the above cases, explicit representations have recently been obtained, independently, by Aldous and Hoover. The aim of the present paper is to continue the work of these authors by deriving some new representations, and by solving the associated uniqueness and continuity problems.
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Research suported by the Air Force Office of Scientific Research Grant No. F 49620 85C 0144
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Kallenberg, O. Some new representations in bivariate exchangeability. Probab. Th. Rel. Fields 77, 415–455 (1988). https://doi.org/10.1007/BF00319298
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DOI: https://doi.org/10.1007/BF00319298