Lattice paths, sampling without replacement, and limiting distributions (Q1028851)

The authors consider lattice paths in the quarter plane \(\mathbb N_0 \times \mathbb N_0\) starting at a point \((m, n)\) in it. Two possible steps are allowed: \((m, n)\to (m - 1, n)\) (to the left) and \((m, n)\to (m, n - 1)\) (downwards). The corresponding weights (probabilities) are equal to \(\frac {m}{m+n}\) and \(\frac {n}{m+n}\), respectively. The authors are interested in paths that cross or touch a certain given line. Such paths are assumed to be absorbed by the line. Let \(Y_{m,n}\) denote the height of the point at which the lattice path touches the line \(y = x/t - s/t\) for the first time. Here \(s\) and \( t, (s, t)\in \mathbb N_0 \times \mathbb N\), are viewed as given paprameters. The sum of the weights of all paths starting at \((m, n)\) for which \[ Y_{m,n} = k \] can be also interpreted as the probability of the latter event \(\mathbb P\{Y_{m,n} = k\}\). The authors obtain an exact expression for this probability. Then, they study the limiting distribution of \(Y_{m,n}\) as \(m \to \infty\) and \(n = n(m)\). Five phase changes are observed. In one of them they obtain the Lévy distribution as a limiting distribution. This implies that the sequence of the moments of the corresponding random variable does not converge. The whole problem is also interpreted in terms of the Pólya-Eggenberger urn model, sampling without replacement and card guessing games.