A note on the asymptotics of lattice paths with general boundaries (Q1923401)

This paper deals with the asymptotic enumeration of lattice paths in \(\mathbb{Z}^2\), with stepsets of the form \({\mathcal S}=\{(1,i)\mid i\in A\}\) for some finite set \(A\), which are constrained to stay between two boundaries \(a\) and \(b\), namely \(a\) and \(b\) are differentiable real valued functions with \(a(0)<0<b(0)\) and \(a(x)<b(x)\) for all \(x>0\), and the numbers \(r(n,k)\) are defined as the number of walks starting from the origin, ending at the point \((n,k)\), which do not cross the boundaries \(a\) and \(b\). In order to obtain an asymptotic evaluation of \(r(n,k)\) as \(n\to\infty\), one considers the lattice paths as realizations of the random walk on \(\mathbb{Z}\), whose increments are equidistributed on the set \(A\). A diffusion approximation of this random walk by a brownian motion with drift can be used to relate the evaluation of \(r(n,k)\) to the corresponding probability for the brownian motion with drift. These probabilities can be obtained by solving some Volterra equations involving the functions \(a\) and \(b\). In general no analytical solutions can be obtained, but these equations are well suited for numerical investigations. Some explicit examples are considered, and it is suggested that the diffusion approximation can be improved by shifting the boundaries in order to take into account the paths which overshoot the boundaries. Some numerical examples of this procedure are shown.