  We prove that the density of
$\frac{X_1+\cdot\cdot\cdot+X_n-nE[X_1]}{\sqrt{n}}$, where $\{X_n\}_{n\geq 1}$
is a sequence of independent and identically distributed random variables
taking values on an abstract Wiener space, converges in $\mathcal{L}^1$ to the
density of a certain Gaussian measure which is absolutely continuous with
respect to the reference Wiener measure. The crucial feature in our
investigation is that we do not require the covariance structure of
$\{X_n\}_{n\geq 1}$ to coincide with the one of the Wiener measure. This
produces a non trivial (different from the constant function one) limiting
object which reflects the different covariance structures involved. The present
paper generalizes the results proved in [18] and deepens the connection between
local limit theorems on (infinite dimensional) Gaussian spaces and some key
tools from the Analysis on the Wiener space, like the Wiener-It\^o chaos
decomposition, Ornstein-Uhlenbeck semigroup and Wick product. We also verify
and discuss our main assumptions on some examples arising from the
applications: dimension independent Berry-Esseen-type bounds and weak solutions
of stochastic differential equations.
