Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Lattice paths, sampling without replacement, and limiting distributions. | Lattice paths sampling without replacement and limiting distributions M. Kuba A. Panholzer and H. Prodinger Institut fur Diskrete Mathematik und Geometrie Technische Universitat Wien Wiedner Hauptstr. 8-10 104 1040 Wien Austria kuba@ and Mathematics Department Stellenbosch University 7602 Stellenbosch South Africa hproding@ Submitted Apr 22 2008 Accepted May 19 2009 Published May 29 2009 Mathematics Subject Classification 05A15 60C05. Abstract In this work we consider weighted lattice paths in the quarter plane No X No. 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 E N and s E No. We provide explicit formula 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 E N and s E No using a generating functions approach. Furthermore these weighted lattice paths can be interpreted as probability distributions arising in the context of Polya-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. Keywords Lattice paths Sampling without replacement urn models Levy distribution This work was supported by the Austrian Science Foundation FWF grant S9608-N23 and by the South African Science Foundation NRF grant 2053748. The second author wants to thank the Department of Mathematical Sciences University of Stellenbosch for its support and hospitality during a research visit where a part of this work has been carried out. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R67 1 1 Introduction Lattice paths Let SC No X No denote a set of lattice points in the quarter plane1. We consider lattice paths with steps m n m 1 n and m n m n 1 .
