  The local (central) limit theorem precisely describes the behavior of
iterated convolution powers of a probability distribution on the
$d$-dimensional integer lattice, $\mathbb{Z}^d$. Under certain mild assumptions
on the distribution, the theorem says that the convolution powers are
well-approximated by a single scaled Gaussian density which we call an
attractor. When such distributions are allowed to take on complex values, their
convolution powers exhibit new and disparate behaviors not seen in the
probabilistic setting. Following works of I. J. Schoenberg, T. N. E. Greville,
P. Diaconis, and L. Saloff-Coste, the author and L. Saloff-Coste provided a
complete description of local limit theorems for the class of finitely
supported complex-valued functions on $\mathbb{Z}$. For convolution powers of
complex-valued functions on $\mathbb{Z}^d$, much less is known. In a previous
work by the author and L. Saloff-Coste, local limit theorems were established
for complex-valued functions whose Fourier transform is maximized in absolute
value at so-called points of positive homogeneous type and, in that case, the
resultant attractors are generalized heat kernels corresponding to a class of
higher order partial differential operators. By considering the possibility
that the Fourier transform can be maximized in absolute value at points of
imaginary homogeneous type, this article extends previous work of the author
and L. Saloff-Coste to broaden the class of complex-valued functions for which
it is possible to obtain local limit theorems. These local limit theorems
contain attractors given by certain oscillatory integrals and their convergence
is established using a generalized polar-coordinate integration formula, due to
H. Bui and the author, and the Van der Corput lemma. The article also extends
recent results on sup-norm type estimates of H. Bui and the author.
