  Suppose $B$ is a Brownian motion and $B^n$ is an approximating sequence of
rescaled random walks on the same probability space converging to $B$ pointwise
in probability. We provide necessary and sufficient conditions for weak and
strong $L^2$-convergence of a discretized Malliavin derivative, a discrete
Skorokhod integral, and discrete analogues of the Clark-Ocone derivative to
their continuous counterparts. Moreover, given a sequence $(X^n)$ of random
variables which admit a chaos decomposition in terms of discrete multiple
Wiener integrals with respect to $B^n$, we derive necessary and sufficient
conditions for strong $L^2$-convergence to a $\sigma(B)$-measurable random
variable $X$ via convergence of the discrete chaos coefficients of $X^n$ to the
continuous chaos coefficients of $X$. In the special case of binary noise, our
results support the known formal analogies between Malliavin calculus on the
Wiener space and Malliavin calculus on the Bernoulli space by rigorous
$L^2$-convergence results.
