Lattice Paths, Sampling Without Replacement, and Limiting Distributions

In this work we consider weighted lattice paths in the quarter plane N0 × N0. 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 t ∈ N and s ∈ N0. 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 t ∈ N and s ∈ N0, 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.