To consider a high-dimensional random process, we propose a notion about stochastic tensor-valued random process (TRP). In this work, we first attempt to apply a generic chaining method to derive tail bounds for all p-th moments of the supremum of TRPs. We first establish tail bounds for suprema of processes with an exponential tail, and further derive tail bounds for suprema of processes with arbitrary number of exponential tails. We apply these bounds to high-dimensional compressed sensing and empirical process characterizations.