Unboundedness of sample functions of stochastic processes with arbitrary parameter sets, with applications to linear and \(l_ p\)-valued parameters (Q791995)

Let X(t), \(t\in T\), be a real valued stochastic process, where T is an arbitrary parameter set. Suppose that there is a nonnegative function d(s,t) on \(T\times T\) such that \(d(s,t)>0\) for \(s\neq t\), and such that the density function of the quotient (X(t)-X(s))/d(s,t) is uniformly smooth for all \(s\neq t\) in the sense that it has an analytic extension to the complex plane which is of specified growth. Then the latter determines a function K(u), \(u>0\), such that if \(\int_{T}\int_{T}K(d(s,t))d\mu(s)d\mu(t)<\infty\) for some measure \(\mu\) with \(\mu(T)>0\), then the sample function is almost surely unbounded on T. The proof is based on the fact that the condition on K implies the existence of an analytic local time for the sample function. The result is applied to \(T=[0,1]\) and \(T=ellipsoid\) in \(l_ p\). The condition of the theorem is shown to be quite sharp in the latter examples.