A combinatorial approach to a model of constrained random walkers

Thibault Espinasse (1) , Nadine Guillotin-Plantard (2, 1) , Philippe Nadeau (2, 3, 4)

1 PSPM - Probabilités, statistique, physique mathématique
2 ICJ - Institut Camille Jordan
3 CTN - Combinatoire, théorie des nombres
4 CNRS - Centre National de la Recherche Scientifique

In [1], the authors consider a random walk $(Z_{n,1},\ldots,Z_{n,K+1})\in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. A functional central limit theorem for the first coordinate is proved and the limit variance is explicited. In this paper, we study an extended version of this model by conditioning the extremal coordinates to be at some fixed distance at every time. We prove a functional central limit theorem for this random walk. Using combinatorial tools, we give a precise formula of the variance and compare it with the one obtained in [1].

Mots clés: Central limit theorem; Martingale; Markov chain

Domaines: Probabilités [math.PR]

Thibault Espinasse, Nadine Guillotin-Plantard, Philippe Nadeau. A combinatorial approach to a model of constrained random walkers. Combinatorics, Probability and Computing, 2016, 25 (2), pp.222-235. ⟨hal-00950845⟩