Learn More
LetM(n), n = 1, 2, . . . be the supercritical branching random walk in which the family sizes may be infinite with positive probability. Assume that a natural martingale related to the M(n) converges almost surely and in mean to a random variable W . For a large subclass of nonnegative and concave functions f we provide a criterion for finiteness of EWf(W(More)
Abstract. Let Xn be the number of cuts needed to isolate the root in a random recursive tree with n vertices. We provide a weak convergence result for Xn. The basic observation for its proof is that the probability distributions of {Xn : n = 2, 3, . . .} are recursively defined by Xn d = Xn−Dn + 1, n = 2, 3, . . ., X1 = 0, where Dn is a discrete random(More)
Given any finite or countable collection of real numbers Tj , j ∈ J , we find all solutions F to the stochastic fixed point equation W d = inf j∈J TjWj , where W and the Wj , j ∈ J , are independent real-valued random variables with distribution F and d = means equality in distribution. The bulk of the necessary analysis is spent on the case when |J | ≥ 2(More)
Abstract We study the total branch length Ln of the Bolthausen-Sznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of Ln are presented. It is shown that Ln/E(Ln) converges to 1 in probability and that Ln, properly normalized, converges weakly to a stable random variable as n tends to infinity. The results are(More)
Partial Quicksort sorts the l smallest elements in a list of length n. We provide a complete running time analysis for this combination of Find and Quicksort. Further we give some optimal adapted versions, called Partition Quicksort, with an asymptotic running time c1l ln l+ c2l+n+o(n). The constant c1 can be as small as the information theoretic lower(More)
Quicksort on the fly returns the input of n reals in increasing natural order during the sorting process. Correctly normalized the running time up to returning the l-th smallest out of n seen as a process in l converges weakly to a limiting process with path in the space of cadlag functions. AMS-classification: Primary 60 F 05, Secondary 68 P 10, 60 K 99
A.Iksanov, S. Polotsky and U. Rösler were supported by the German Research Foundation (project no. 436UKR 113/93/0-1). The research leading to the present paper has been mainly conducted during visits to University of Kiev (Rösler), to University of Kiel (Iksanov and Polotsky), and to University of Münster (Iksanov). Financial support obtained from these(More)