Limit Laws for Basic Parameters of Lattice Paths with Unbounded Jumps

Abstract

This paper establishes the asymptotics of a class of random walks on ℕ with regular but unbounded jumps and studies several basic parameters (returns to zero for meanders, bridges, excursions, final altitude for meanders). All these results are generic (obtained by the kernel method for the combinatorial part and by singularity analysis for the asymptotic part).This paper completes the article [3] which was only dealing with the combinatorics (enumeration and bijections) of walks with unbounded jumps (the so-called “factorial walks”), which play an important role for uniform random generation of some combinatorial objects. We fully parallelize the analytical approach from [4] which was dealing with walks with bounded jumps only.KeywordsKernel MethodDiscrete MathematicLaurent SeriesProbability Generate FunctionLattice PathThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.