This paper investigates the asymptotic distribution of order statistics in generalized allocation schemes. We focus on scenarios where n, the number of particles, and N, the number of cells, satisfy \(n = \theta N^a\) for some \(a > 1\) and a positive constant \(\theta \) . Building on foundational work by Kolchin and others, where the joint distribution of the number of particles in each cell, \((\eta _1 = k_1, \ldots , \eta _N = k_N)\) , follows a conditional joint distribution \((\xi _1 = k_1, \ldots , \xi _N = k_N \mid \xi _1 + \cdots + \xi _N = n)\) , we assume that \(\xi _i\) are independent geometric random variables and \(k_1 + \cdots + k_N = n\) . We extend the analysis to determine the limiting distributions and the moment generating functions of the order statistics \(\eta _{(1)}\le \eta _{(2)}\le \cdots \le \eta _{(N)}\) under these allocation models. We present the limiting distributions for \(\eta _{(m)}\) and the moment generating functions of \(\eta _{(m)} / N^{a-2}\) for \(1\le m \le N\) under certain conditions. These findings have significant implications for applications in random graphs and sequences of runs, providing deeper insights into their asymptotic behavior.
