  Smooth linearization is one of the central themes in the study of dynamical
systems. The classical Belitskii's $C^1$ linearization theorem has been widely
used in the investigation of dynamical behaviors such as bifurcations, mixing,
and chaotic behaviors due to its minimal requirement of partial second order
non-resonances and low regularity of systems. In this article, we revisit
Belitskii's $C^1$ linearization theorem by taking an approach based on smooth
invariant foliations and study this problem for a larger class of dynamical
systems ({\it random dynamical systems}). We assumed that the linearized system
satisfies the condition of Multiplicative Ergodic Theorem and the associated
Lyapunov exponents satisfy Belitskii's partial second order non-resonant
conditions. We first establish the existence of $C^{1,\beta}$ stable and
unstable foliations without assuming the bunching condition for Lyapunov
exponents, then prove a $C^{1,\beta}$ linearization theorem of Belitskii type
for random dynamical systems. As a result, we show that the classical
Belitskii's $C^1$ linearization theorem for a $C^{2}$ diffeomorphism $F$ indeed
holds without assuming all eigenspaces of the linear system $DF(0)$ are
invariant under the nonlinear system $F$, a requirement previously imposed by
Belitskii in his proof.
