  We consider the Wiener integral with respect to a $d$-parameter Hermite
process with Hurst multi-index ${\bf H}= (H_{1},\ldots, H_{d}) \in \left(
\frac{1}{2}, 1\right) ^{d}$ and we analyze the limit behavior in distribution
of this object when the components of ${\bf H}$ tend to $1$ and/or
$\frac{1}{2}$. As examples, we focus on the solution to the stochastic heat
equation with additive Hermite noise and to the Hermite Ornstein-Uhlenbeck
process.
