  In this paper, we propose a kernel method for exact tail asymptotics of a
random walk to neighborhoods in the quarter plane. This is a two-dimensional
method, which does not require a determination of the unknown generating
function(s). Instead, in terms of the asymptotic analysis and a Tauberian-like
theorem, we show that the information about the location of the dominant
singularity or singularities and the detailed asymptotic property at a dominant
singularity is sufficient for the exact tail asymptotic behaviour for the
marginal distributions and also for joint probabilities along a coordinate
direction. We provide all details, not only for a "typical" case, the case with
a single dominant singularity for an unknown generating function, but also for
all non-typical cases which have not been studied before. A total of four types
of exact tail asymptotics are found for the typical case, which have been
reported in the literature. We also show that on the circle of convergence, an
unknown generating function could have two dominant singularities instead of
one, which can lead to a new periodic phenomena. Examples are illustrated by
using this kernel method. This paper can be considered as a systematic summary
and extension of existing ideas, which also contains new and interesting
research results.
