The covariance structure of random permutation matrices
Marlos A. G. Viana · Contemporary mathematics - American Mathematical Society · 2001
Random permutation matrices (U) arise naturally in the stochas- tic representation of vectors of order statistics, induced order statistics and associated ranks. When the probability law of U is uniform, the covariance structure among the entries of U is derived explicitely, and a constructive derivation for the covariance in the general case (U k ) is described and related to the cyclic structure of the symmetric group. It is shown that the covari- ance structure of vectors resulting from the multiplicative action UX of U on a given vector X requires averaging the symmetric conjugates UXX 0 U 0 of XX' over the group of permutation matrices. The mean conjugate is a projection operator which leads to a trace decomposition with the usual ANOVA inter- pretation. Numerical examples of these decompositions are discussed in the context of the analysis of circularly symmetric data.