Principal Component Analysis for a Stationary Random Function Defined on a Locally Compact Abelian Group

When Z is a random L2H-valued measure, where H is a Hilbert space, we prove that there exists an L2q-valued measure, which may depend on constraints and which best sums up the random measure Z according to a stationary criterion. Then a technique to reduce a random function is deduced from the above result. The random function is defined on a locally compact abelian group and is stationary and continuous. This work generalizes Brillinger's results on stationary time series.