On the Supremum of Certain Families of Stochastic Processes

We consider a family of stochastic processes $\{X_t^{ϵ},t\in T\}$ on a metric space $T$, with a parameter $ϵ\downarrow 0$. We study the conditions under which \lim_{\e \to 0} \P \Big(\sup_{t \in T} |X_t^\e| < \delta \Big) =1 when one has the $\mathit{\text{a}}\mathit{\text{priori}}$ estimate on the modulus of continuity and the value at one point. We compare our problem to the celebrated Kolmogorov continuity criteria for stochastic processes, and finally give an application of our main result for stochastic intergrals with respect to compound Poisson random measures with infinite intensity measures.