We consider a Markov chain (M n )n≥0 on the set ℕ0 of nonnegative integers which is eventually decreasing, i.e. ℙ{M n+1<M n   |  M n ≥a}=1 for some a∈ℕ and all n≥0. We are interested in the asymptotic behavior of the law of the stopping time T=T(a)≔inf{k∈ℕ0:  M k <a} under ℙn ≔ℙ (·  |  M 0=n) as n→∞. Assuming that the decrements of (M n )n≥0 given M 0=n possess a kind of stationarity for large n, we derive sufficient conditions for the convergence in the minimal L p -distance of ℙn (T−a n)∕b n∈·) to some nondegenerate, proper law and give an explicit form of the constants a n and b n .

We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1-stable law, as the initial number of particles goes to ∞. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b = 1, which corresponds to the Bolthausen-Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta (a, b)-coalescents with 0 < a < 1 leads to a simplified derivation of the known (2 - a)-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta (1, b)-coalescent by exploiting the method of sequential approximations.

We consider the Λ-coalescent processes with a positive frequency of singleton clusters. The class in focus covers, for instance, the beta(a, b)-coalescents with a > 1. We show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing Lévy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of collisions.

We study the number of collisions, Xn, of an exchangeable coalescent with multiple collisions (Λ-coalescent) which starts with n particles and is driven by rates determined by a finite characteristic measure η(dx) = x−2Λ(dx). Via a coupling technique, we derive limiting laws of Xn, using previous results on regenerative compositions derived from stick-breaking partitions of the unit interval. The possible limiting laws of Xn include normal, stable with index 1 ≤ α < 2, and Mittag-Leffler distributions. The results apply, in particular, to the case when η is a beta(a − 2, b) distribution with parameters a > 2 and b > 0. The approach taken allows us to derive asymptotics of three other functionals of the coalescent: the absorption time, the length of an external branch chosen at random from the n external branches, and the number of collision events that occur before the randomly selected external branch coalesces with one of its neighbours.

Let S0 := 0 and Sk := ξ1 + ··· + ξk for k ∈ ℕ := {1, 2, …}, where {ξk : k ∈ ℕ} are independent copies of a random variable ξ with values in ℕ and distribution pk := P{ξ = k}, k ∈ ℕ. We interpret the random walk {Sk : k = 0, 1, 2, …} as a particle jumping to the right through integer positions. Fix n ∈ ℕ and modify the process by requiring that the particle is bumped back to its current state each time a jump would bring the particle to a state larger than or equal to n. This constraint defines an increasing Markov chain {Rk(n) : k = 0, 1, 2, …} which never reaches the state n. We call this process a random walk with barrier n. Let Mn denote the number of jumps of the random walk with barrier n. This paper focuses on the asymptotics of Mn as n tends to ∞. A key observation is that, under p1 > 0, {Mn : n ∈ ℕ} satisfies the distributional recursion M1 = 0 and for n = 2, 3, …, where In is independent of M2, …, Mn−1 with distribution P{In = k} = pk / (p1 + ··· + pn−1), k ∈ {1, …, n − 1}. Depending on the tail behavior of the distribution of ξ, several scalings for Mn and corresponding limiting distributions come into play, including stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is to compare (couple) the number of jumps, Mn, with the first time, Nn, when the unrestricted random walk {Sk : k = 0, 1, …} reaches a state larger than or equal to n. The results are applied to derive the asymptotics of the number of collision events (that take place until there is just a single block) for β(a, b)-coalescent processes with parameters 0 < a < 2 and b = 1.

A class of haploid population models with population size N, nonoverlapping generations and exchangeable offspring distribution is considered. Based on an analysis of the discrete ancestral process, we present solutions, algorithms and strong upper bounds for the expected time back to the most recent common ancestor which hold for arbitrary sample size n ∈ {1,…,N}. New insights into the asymptotic behaviour of the expected time back to the most recent common ancestor for large population size are presented relating the results to coalescent theory.

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift.

Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.