Bounded discrete walks

This article tackles the enumeration and asymptotics of directed lattice paths (that are isomorphic to unidimensional paths) of bounded height (walks below one wall, or between two walls, for any finite set of jumps). Thus, for any lattice paths, we give the generating functions of bridges (“discrete” Brownian bridges) and reflected bridges (“discrete” reflected Brownian bridges) of a given height. It is a new success of the “kernel method ” that the generating functions of such walks have some nice expressions as symmetric functions in terms of the roots of the kernel. These formulae also lead to fast algorithms for computing the n-th Taylor coefficients of the corresponding generating functions. For a large class of walks, we give the discrete distribution of the height of bridges, and show the convergence to a Rayleigh limit law. For the family of walks consisting of a −1 jump and many positive jumps, we give more precise bounds for the speed of convergence. We end our article with a heuristic application to bioinformatics that has a high speed-up relative to previous work

Bounded discrete walks

This article tackles the enumeration and asymptotics of directed lattice paths (that are isomorphic to unidimensional paths) of bounded height (walks below one wall, or between two walls, for $\textit{any}$ finite set of jumps). Thus, for any lattice paths, we give the generating functions of bridges ("discrete'' Brownian bridges) and reflected bridges ("discrete'' reflected Brownian bridges) of a given height. It is a new success of the "kernel method'' that the generating functions of such walks have some nice expressions as symmetric functions in terms of the roots of the kernel. These formulae also lead to fast algorithms for computing the $n$-th Taylor coefficients of the corresponding generating functions. For a large class of walks, we give the discrete distribution of the height of bridges, and show the convergence to a Rayleigh limit law. For the family of walks consisting of a $-1$ jump and many positive jumps, we give more precise bounds for the speed of convergence. We end our article with a heuristic application to bioinformatics that has a high speed-up relative to previous work

Constant time estimation of ranking statistics by analytic combinatorics

We consider i.i.d. increments (or jumps) Xi that are integers in J ⊆ [−c,..., +d] for c, d ∈ N, the partial sums Sj = P 1≤i≤j Xi, and the discrete walks ((j, Sj))1≤j≤n. Late conditionning by a return of the walk to zero at time n provides discrete bridges that we note (Bj)1≤j≤n. We give in this extended abstract the asymptotic law in the central domain of the height (max1≤j≤n Bj) of the bridges as n tends to infinity. As expected, this law converges to the Rayleigh law which is the law of the maximum of a standard Brownian bridge. In the case where c = 1 (only one negative jump), we provide a full expansion of the asymptotic limit which improves upon the rate of convergence O(log(n) / √ n) given by Borisov [4] for lattice jumps; this applies in particular for the case where Xi ∈ {−1, +d}, in which case the expansion is expressible as a function of n, d and of the height of the bridge. Applying this expansion for Xi ∈ {−1, d/c} gives an excellent approximation of the case Xi ∈ {−d, +c} and provides in constant time an indicator used in ranking statistics; this indicator can be used for medical diagnosis and bioinformatics analysis (see Keller et al. [7] who compute it in time O(n × min(c, d)) by use of dynamical programming)