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This paper deals with asymptotics for multiple-set linear canonical analysis (MSLCA). A definition of this analysis, that adapts the classical one to the context of Euclidean random variables, is given and properties of the related canonical coefficients are derived. Then, estimators of the MSLCA's elements, based on empirical covariance operators, are proposed and asymptotics for these estimators are obtained. More precisely, we prove their consistency and we obtain asymptotic normality for the estimator of the operator that gives MSLCA, and also for the estimator of the vector of canonical coefficients. These results are then used to obtain a test for mutual non-correlation between the involved Euclidean random variables.

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Measures of association are introduced for Hilbertian subspaces, that are defined by a few axioms and are shown to be symmetric nondecreasing functions of the canonical coefficients. When particular subspaces are considered, classical measures of association are obtained as particular cases. Moreover, the proposed framework allows one to introduce new approaches for measuring partial noncorrelation, partial independence and linear predictability of a stationary process.

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