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Selfsimilar scaling limits of nonincreasing Markov chains
, 2009
"... We study scaling limits of nonincreasing Markov chains with values in the set of nonnegative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from n and appropriately resca ..."
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Cited by 11 (1 self)
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We study scaling limits of nonincreasing Markov chains with values in the set of nonnegative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from n and appropriately rescaled, converges in distribution, as n → ∞, to a nonincreasing selfsimilar Markov process. This convergence holds jointly with that of the rescaled absorption time to the time at which the selfsimilar Markov process reaches first 0. We discuss various applications to the study of random walks with a barrier, of the number of collisions in Λcoalescents that do not descend from infinity and of nonconsistent regenerative compositions. Further applications to the scaling limits of Markov branching trees are developed in the forthcoming paper [11].
ON THE NUMBER OF ALLELIC TYPES FOR SAMPLES TAKEN FROM EXCHANGEABLE COALESCENTS WITH MUTATION
, 2008
"... Let Kn denote the number of types of a sample of size n taken from an exchangeable coalescent process (Ξcoalescent) with mutation. A distributional recursion for the sequence (Kn)n∈N is derived. If the coalescent does not have proper frequencies, i.e., if the characterizing measure Ξ on the infinit ..."
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Cited by 2 (1 self)
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Let Kn denote the number of types of a sample of size n taken from an exchangeable coalescent process (Ξcoalescent) with mutation. A distributional recursion for the sequence (Kn)n∈N is derived. If the coalescent does not have proper frequencies, i.e., if the characterizing measure Ξ on the infinite simplex ∆ does not have mass at zero and satisfies R ∆ xΞ(dx)/(x, x) < ∞, where x : = P∞ i=1 xi and (x,x): = P∞
On the asymptotics of moments of linear random recurrences
 Theory Stoch. Proc
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unknown title
, 2010
"... We propose a new method of analyzing the asymptotics of moments of certain linear random recurrences which is based on the technique of iterative functions. By using the method, we show that the moments of the number of collisions and the absorption time in the PoissonDirichlet coalescent behave l ..."
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We propose a new method of analyzing the asymptotics of moments of certain linear random recurrences which is based on the technique of iterative functions. By using the method, we show that the moments of the number of collisions and the absorption time in the PoissonDirichlet coalescent behave like the powers of the "log star" function which grows slower than any iteration of the logarithm, and thereby we prove a weak law of large numbers. Finally, we discuss merits and limitations of the method and give several examples related to beta coalescents, recursive algorithms, and random trees. Introduction and main result A linear random recurrence is a sequence of random variables {X n , n ∈ N} which satisfies the distributional equality where X n is some parameter of a problem of size n which splits into K ≥ 1 subproblems of random sizes I n (r) ∈ {1, . . . , n}. For every r = 1, . . . , K, the sequence {X (r) k , k ∈ N} which corresponds to the contribution of subgroup r is a distributional copy of {X k , k ∈ N}, V n is a random toll term, and A r (n) > 0 is a random weight of subgroup r. It is assumed that Random recurrences (1), often in a simplified form with K = 1, arise in various areas of applied probability such as random regenerative structures The first step of the asymptotic analysis of recurrences where {b n , n ∈ N} and {c nk , n ∈ N, k < n} are given numerical sequences. The purpose of the present paper is to propose a new method of obtaining the firstorder asymptotics of solutions to (2), as n → ∞. Although the asymptotic analysis of recurrences (2) is a hard analytic problem, some more or less efficient methods have been elaborated to date. Evidently, the most popular existing approach is the method of singular analysis of generating functions