  In this work, we examine the optimality of Gaussian signalling for covert
communications with an upper bound on $\mathcal{D}(p_{_1}||p_{_0})$ or
$\mathcal{D}(p_{_0}||p_{_1})$ as the covertness constraint, where
$\mathcal{D}(p_{_1}||p_{_0})$ and $\mathcal{D}(p_{_0}||p_{_1})$ are different
due to the asymmetry of Kullback-Leibler divergence, $p_{_0}(y)$ and
$p_{_1}(y)$ are the likelihood functions of the observation ${y}$ at the warden
under the null hypothesis (no covert transmission) and alternative hypothesis
(a covert transmission occurs), respectively. Considering additive white
Gaussian noise at both the receiver and the warden, we prove that Gaussian
signalling is optimal in terms of maximizing the mutual information of
transmitted and received signals for covert communications with an upper bound
on $\mathcal{D}(p_{_1}||p_{_0})$ as the constraint. More interestingly, we also
prove that Gaussian signalling is not optimal for covert communications with an
upper bound on $\mathcal{D}(p_{_0}||p_{_1})$ as the constraint, for which as we
explicitly show skew-normal signalling can outperform Gaussian signalling in
terms of achieving higher mutual information. Finally, we prove that, for
Gaussian signalling, an upper bound on $\mathcal{D}(p_{_1}||p_{_0})$ is a
tighter covertness constraint in terms of leading to lower mutual information
than the same upper bound on $\mathcal{D}(p_{_0}||p_{_1})$, by proving
$\mathcal{D}(p_{_0}||p_{_1}) \leq \mathcal{D}(p_{_1}||p_{_0})$.
