Criteria for the Continuity and Differentiability of Vector Parameter Processes

摘要

Let f(x) be a real or vector valued random process whose parameter set R is a bounded set in Euclidean n-space. Following the usual terminology, a version of f(x) is any process f tilde (x) defined on R which satisfies P(f(x) = f tilde (x)) =1 for each x epsilon R. It is sometimes useful to assert that there exists a version of f(x) which is (with probability one) continuous, Holder continuous, or perhaps differentiable in some component. Also estimates of the type P(sup (x epsilon R)/f(x)/> epsilon) may be desired for these versions. Illustrative examples are given.