Occupancy schemes associated to Yule processes

An occupancy problem with an infinite number of bins and a random probability vector for the locations of the balls is considered. The respective sizes of the bins are related to the split times of a Yule process. The asymptotic behavior of the landscape of the first empty bins, i.e. the set of corresponding indices represented by point processes, is analyzed and convergences in distribution to mixed Poisson processes are established. Additionally, the influence of the random environment, the random probability vector, is analyzed. It is represented by two main components: an independent, identically distributed sequence and a fixed random variable. Each of these components has a specific impact on the qualitative behavior of the stochastic model. It is shown in particular that, for some values of the parameters, some rare events, which are identified, determine the asymptotic behavior of the average values of the number of empty bins in some regions.

[1]  Donald A. Dawson,et al.  Measure-valued Markov processes , 1993 .

[2]  S. Karlin.  Central Limit Theorems for Certain Infinite Urn Schemes , 1967 .

[3]  N. L. Johnson,et al.  Urn models and their application : an approach to modern discrete probability theory , 1978 .

[4]  Olle Nerman,et al.  On the convergence of supercritical general (C-M-J) branching processes , 1981 .

[5]  M. Wodzicki.  Lecture Notes in Math , 1984 .

[6]  J. Pitman,et al.  Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws ∗ , 2007, math/0701718.

[7]  J. Neveu,et al.  Martingales à temps discret , 1973 .

[8]  Philippe Robert.  Stochastic Networks and Queues , 2003 .

[9]  A. Gnedin.  Bernoulli Sieve , 2003, math/0303071.

[10]  D. A. Sprott.  Urn Models and Their Application—An Approach to Modern Discrete Probability Theory , 1978 .

[11]  M. Härnqvist.  Limit theorems for point processes generated in a general branching process , 1981, Advances in Applied Probability.

[12]  A. Gnedin,et al.  Small parts in the Bernoulli sieve , 2008, 0804.3052.

[13]  Philippe Robert,et al.  A queueing system for modeling a file sharing principle , 2008, SIGMETRICS '08.

[14]  Philippe Flajolet,et al.  Probabilistic Counting Algorithms for Data Base Applications , 1985, J. Comput. Syst. Sci..

[15]  Norman L. Johnson,et al.  Urn models and their application , 1977 .

[16]  K. Athreya,et al.  Convergence of the Age Distribution in the One-Dimensional Supercritical Age-Dependent Branching Process , 1976 .

[17]  J. Kingman.  The First Birth Problem for an Age-dependent Branching Process , 1975 .

[18]  David Williams,et al.  Probability with Martingales , 1991, Cambridge mathematical textbooks.

[19]  Mikko Alava,et al.  Branching Processes , 2009, Encyclopedia of Complexity and Systems Science.

[20]  Svante Janson,et al.  LOCAL LIMIT THEOREMS FOR FINITE AND INFINITE URN MODELS , 2006, math/0604397.

[21]  Quansheng Liu,et al.  Asymptotic properties and absolute continuity of laws stable by random weighted mean , 2001 .

[22]  Fabrice Guillemin,et al.  Measurement Based Modeling of eDonkey Peer-to-Peer File Sharing System , 2007, International Teletraffic Congress.

[23]  R. Cooke.  Real and Complex Analysis , 2011 .

[24]  A. Barbour,et al.  Poisson Approximation , 1992 .