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2 results

  • Large deviations for the leaves in some random trees

  • Part of
  • Wlodek Bryc, David Minda, Sunder Sethuraman
  • Journal:
    Advances in Applied Probability / Volume 41 / Issue 3 / September 2009
    Published online by Cambridge University Press:
    01 July 2016, pp. 845-873
    Print publication:
    September 2009
    • Article
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  • Large deviation principles and related results are given for a class of Markov chains associated to the ‘leaves' in random recursive trees and preferential attachment random graphs, as well as the ‘cherries’ in Yule trees. In particular, the method of proof, combining analytic and Dupuis–Ellis-type path arguments, allows for an explicit computation of the large deviation pressure.

  • The asymptotic behaviour of birth and death and some related processes

  • Andrew D. Barbour
  • Journal:
    Advances in Applied Probability / Volume 7 / Issue 1 / March 1975
    Published online by Cambridge University Press:
    01 July 2016, pp. 28-43
    Print publication:
    March 1975

  • The paper examines those continuous time Markov processes Z(·) on the positive integers which have the ‘skip free upwards’ property, with regard to their asymptotic behaviour in the event of Z(t) tending to infinity. The behaviour is characterised in terms of the convergence or divergence of an appropriate function of Z(t), and the description is improved by central limit and iterated logarithm theorems. The conditions of the theorems are expressed entirely in terms of the matrix Q of instantaneous transition rates for Z(·). The method is applied, by way of example, to the super-critical linear birth and death process.