Hölder Continuity of Random Processes

For a Young function φ and a Borel probability measure m on a compact metric space (T,d) the minorizing metric is defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau_{m,\varphi}(s,t):=\max\biggl\{\int^{d(s,t)}_{0}\varphi^{-1}\biggl(\frac{1}{m(B(s,\varepsilon))}\biggr)d\varepsilon,\int^{d(s,t)}_{0}\varphi^{-1}\biggl(\frac{1}{m(B(t,\varepsilon ))}\biggr)d\varepsilon\biggr\}.$$\end{document} In the paper we extend the result of Kwapien and Rosinski (Progr. Probab. 58, 155–163, 2004) relaxing the conditions on φ under which there exists a constant K such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{E}\sup_{s,t\in T}\varphi\biggl(\frac{|X(s)-X(t)|}{K\tau _{m,\varphi}(s,t)}\biggr)\leq 1,$$\end{document} for each separable process X(t), t∈T which satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sup_{s,t\in T}\mathbf{E}\varphi(\frac {|X(s)-f(t)|}{d(s,t)})\leq 1$\end{document} . In the case of φp(x)≡xp, p≥1 we obtain the somewhat weaker results.