Lattice Paths, Sampling Without Replacement, and Limiting Distributions - Kuba, M.; Panholzer, A.; Prodinger, H. (The Electronic Journal of Combinatorics, 2009)
In this work we consider weighted lattice paths in the quarter plane N_0×N_0. The steps are given by (m,n)→(m-1,n), (m,n)→(m,n-1) and are weighted as follows: (m,n)→(m-1,n) by m/(m+n) and step (m,n)→(m,n-1) by n/(m+n). The considered lattice paths are absorbed at lines y=x/t -s/t with tN and sN_0. We provide explicit formulæ for the sum of the weights of paths, starting at (m,n), which are absorbed at a certain height k at lines y=x/t -s/t with tN and sN_0, using a generating functions approach. Furthermore these weighted lattice paths can be interpreted as probability distributions arising in the context of Pólya-Eggenberger urn models, more precisely, the lattice paths are sample paths of the well known sampling without replacement urn. We provide limiting distribution results for the underlying random variable, obtaining a total of five phase changes.