Stochastic Processes with Value in Exponential Type Orlicz Spaces

Abstract

Let $(T, \Theta)$ be a compact measurable topological space and $\Psi_q(x) = \exp|x|^q - 1, 1 \leq q < \infty$. Let $X = \{X(\omega, t), \omega \in \Omega, t \in T\}$ be a $\Theta$-measurable stochastic process such that $\|X(s) - X(t)\|_{L^\Psi q(\Omega)}\leq d(s, t)$ for every $(s, t) \in T \otimes T$, where $d(\cdot, \cdot)$ is some continuous pseudometric on $(T, \Theta)$. We give a sufficient condition expressed in terms of a majorizing measure on $(T, d)$ in order that $X$ take values in the Orlicz space $L^{\Psi_q}(T, \mu)$, where $q \leq q' < \infty$ and $\mu$ any Borel probability measure on $(T, \Theta)$.