Sample Boundedness of Stochastic Processes Under Increment Conditions

Abstract

Let $(T, d)$ be a compact metric space of diameter $D$, and $\|\cdot \|_\Phi$ be an Orlicz norm. When is it true that all (separable) processes $(X_t)_{t \in T}$ that satisfy the increment condition $\|X_t - X_s\|_\Phi \leq d(t, s)$ for all $s, t$ in $T$ are sample bounded? We give optimal necessary conditions and optimal sufficient conditions in terms of the existence of a probability measure $m$ on $T$ that satisfies an integral condition $\int^D_0 f(\varepsilon, m(B(x, \varepsilon))) d\varepsilon \leq K$ for each $x$ in $T$, where $f$ is a function suitably related to $\Phi$. When $T$ is a compact group and $d$ is translation invariant, we are able to compute the necessary and sufficient condition in several cases.