Dilations of $V$-bounded stochastic processes indexed by a locally compact group

Abstract:

It is proved that a stochastic process (i.e., a Hilbert space valued function) indexed by a locally compact group is $V$-bounded (i.e., weakly harmonizable in an appropriate sense) if, and only if, it can be expressed as an orthogonal projection of a process whose covariance function $R$ satisfies $R(s,t) = \rho ({t^{ - 1}}s) + \rho (s{t^{ - 1}})$ for some continuous positive-definite function $\rho$. The result generalizes a well-known theorem due to H. Niemi, and depends on the noncommutative Grothendieck type inequality of G. Pisier.