Hölder Continuity of Random Processes

Abstract

For a Young function φ and a Borel probability measure m on a compact metric space (T,d) the minorizing metric is defined by

$$\tau_{m,\varphi}(s,t):=\max\biggl\{\int^{d(s,t)}_{0}\varphi^{-1}\biggl(\frac{1}{m(B(s,\varepsilon))}\biggr)d\varepsilon,\int^{d(s,t)}_{0}\varphi^{-1}\biggl(\frac{1}{m(B(t,\varepsilon ))}\biggr)d\varepsilon\biggr\}.$$

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

$$\mathbf{E}\sup_{s,t\in T}\varphi\biggl(\frac{|X(s)-X(t)|}{K\tau _{m,\varphi}(s,t)}\biggr)\leq 1,$$

for each separable process X(t), t∈T which satisfies $\sup_{s,t\in T}\mathbf{E}\varphi(\frac {|X(s)-f(t)|}{d(s,t)})\leq 1$ . In the case of φ _p (x)≡x ^p, p≥1 we obtain the somewhat weaker results.