Linglong Yuan

On the total length of external branches for beta-coalescents

In this paper we consider the beta(2 − α, α)-coalescents with 1 < α < 2 and study the moments of external branches, in particular, the total external branch length of an initial sample of n individuals. For this class of coalescents, it has been proved that n α-1 T (n) →D T, where T (n) is the length of an external branch chosen at random and T is a known nonnegative random variable. For beta(2 − α, α)-coalescents with 1 < α < 2, we obtain lim n→+∞ n 3α-5 𝔼(L ext (n) − n 2-α𝔼T)2 = ((α − 1)Γ(α + 1))2Γ(4 − α) / ((3 − α)Γ(4 − 2α)).

A generalization of Kingman’s model of selection and mutation and the Lenski experiment

Kingmans model of selection and mutation studies the limit type value distribution in an asexual population of discrete generations and infinite size undergoing selection and mutation. This paper generalizes the model to analyze the long-term evolution of Escherichia. coli in Lenski experiment. Weak assumptions for fitness functions are proposed and the mutation mechanism is the same as in Kingmans model. General macroscopic epistasis are designable through fitness functions. Convergence to the unique limit type distribution is obtained.

A Note on the Small-Time Behaviour of the Largest Block Size of Beta n -Coalescents

We study the largest block size of Beta n-coalescents at small times as n tends to infinity, using the paintbox construction of Beta-coalescents and the link between continuous-state branching processes and Beta-coalescents established in Birkner et al. (Electron J Probab 10(9):303–325, 2005) and Berestycki et al. (Ann Inst H Poincare Probab Stat 44(2):214–238, 2008). As a corollary, a limit result on the largest block size at the coalescence time of the individual/block {1} is provided.

A probabilistic interpretation of the Gaussian binomial coefficients

We present a stand-alone simple proof of a probabilistic interpretation of the Gaussian binomial coefficients by conditioning a random walk to hit a given lattice point at a given time.