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In this paper, we propose a kernel method for exact tail asymptotics of arandom walk to neighborhoods in the quarter plane. This is a two-dimensionalmethod, which does not require a determination of the unknown generatingfunction(s). Instead, in terms of the asymptotic analysis and a Tauberian-liketheorem, we show that the information about the location of the dominantsingularity or singularities and the detailed asymptotic property at a dominantsingularity is sufficient for the exact tail asymptotic behaviour for themarginal distributions and also for joint probabilities along a coordinatedirection. We provide all details, not only for a "typical" case, the case witha single dominant singularity for an unknown generating function, but also forall non-typical cases which have not been studied before. A total of four typesof exact tail asymptotics are found for the typical case, which have beenreported in the literature. We also show that on the circle of convergence, anunknown generating function could have two dominant singularities instead ofone, which can lead to a new periodic phenomena. Examples are illustrated byusing this kernel method. This paper can be considered as a systematic summaryand extension of existing ideas, which also contains new and interestingresearch results.

A Kernel Method for Exact Tail Asymptotics --- Random Walks in the Quarter Plane