  In this paper, we obtain a $p$-th moment bound for the suprema of a
log-concave-tailed nonhomogeneous chaos process, which is optimal in some
special cases. A crucial ingredient of the proof is a novel decoupling
inequality, which may be of independent interest. With this $p$-th moment
bound, we show two uniform Hanson-Wright type deviation inequalities for
$\alpha$-subexponential entries ($1\le \alpha\le 2$), which recover some known
results. As applications, we prove the restricted isometry property of partial
random circulant matrices and time-frequency structured random matrices induced
by standard $\alpha$-subexponential vectors ($1\le \alpha\le 2$), which extends
the previously known results for the subgaussian case.
