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...has been used in this paper. In the lattice case, when X1 assumes only positive integer values, the whole range of possible distributional limits follows from Theorems 1.2, 1.5 and Proposition 3.1 in =-=[21]-=-. Although the first two of these results were formulated for other variables, they apply to Nt as well. As in [21], the same asymptotic results are readily extendible to the first passage time proces...

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... mean rLn for some fixed mutation rate r > 0. Our first main new contribution is the proof of a 1-stable limit law for Xn and Ln as n → ∞. As in much of the previous work (see, for instance, [14] and =-=[20]-=-) we use a renewal approximation to Πn. A novel element in this context is estimating the quality of approximation in terms of a Wasserstein distance. The second new contribution are asymptotic expans...

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...able ξ taking values in the set of multiples of 100 kg. Then T10001, M10001 and V10001 define the total number of requests, the number of requests to be satisfied and to be rejected, respectively. In =-=[5]-=- and [7] the weak convergence of properly normalized and centered Mn and Tn, respectively, was investigated. Assuming that the distribution of ξ belongs to the domain of attraction of a stable law, a ...

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...ent event. The construction using discrete trees allows us to recover in Section 4.1 the asymptotic distribution of the number of coalescent events given by [19] in a more general framework, see also =-=[21]-=-. Proposition 1.2. Let X ′ n be the number of collisions undergone by (Π[n](t),t ≥ 0). Then we have: X ′ n √ n (d) √ −→ 2Z, n→∞ where Z has a Rayleigh distribution with density xe −x2 /2 1{x>0}. Let u...

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... and stay infinite for a ≥ 1. With the account of results of the present paper, we have the following list. (i) Case 0 < a < 1. The limit law of (Xn− (1− a)n)/n1/(2−a) is (2− a)-stable (see [16], and =-=[21]-=- for the case b = 1). The distribution of τ∞ is unknown. (ii) Case a = 1. The instance b = 1 is the Bolthausen-Sznitman coalescent, for which the limit distribution of τn− log log n is standard Gumbel...

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...res [9, 11], random trees [5, 7, 25, 26], coalescent theory [6, 10, 12, 16], absorption times in non-increasing Markov chains [13, 2], recursive algorithms [15, 24, 27, 28], random walks with barrier =-=[17, 18]-=-, to name but a few. The first step of asymptotic analysis of recurrences (1) is to find the asymptotics of moments EXkn and central moments E(Xn − EXn) k, as n → ∞. This problem reduces to studying t...

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...ith no killing and the Lévy measure να(dt) = e−t/α (1− e−t/α)α+1 1(0,∞)(t)dt. One can check that Φα(x) := − logEe −xYα(1) = Γ(1− α)Γ(αx+ 1) Γ(α(x− 1) + 1) − 1, x ≥ 0. It is known (see, for instance, =-=[13]-=-) that X←α (1) d = ∫ T 0 e−Yα(t)dt. and that X←α (1) has a Mittag-Leffler law which is uniquely determined by its moments E(X←α (1)) n = n! (Φα(1) + 1) . . . (Φα(n) + 1) = n! Γ(1 + nα)Γn(1− α) , n ∈ N...