The Cannings exchangeable model for a finite population in discrete time is extended to incorporate selection. The probability of fixation of a mutant type is studied under the assumption of weak selection. An exact formula for the derivative of this probability with respect to the intensity of selection is deduced, and developed in the case of a single mutant. This formula is expressed in terms of mean coalescence times under neutrality assuming that the coefficient of selection for the mutant type has a derivative with respect to the intensity of selection that takes a polynomial form with respect to the frequency of the mutant type. An approximation is obtained in the case where this derivative is a continuous function of the mutant frequency and the population size is large. This approximation is consistent with a diffusion approximation under moment conditions on the number of descendants of a single individual in one time step. Applications to evolutionary game theory in finite populations are presented.

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Abstract. Let (Sn) n≥0 be a Z-random walk and (ξx) x∈Z be a sequence of independent and identically distributed R-valued random variables, independent of the random walk. Let h be a measurable, symmetric function defined on R 2 with values in R. We study the weak convergence of the sequence Un, n ∈ N, with values in D[0, 1] the set of right continuous real-valued functions with left limits, defined by [nt] i,j=0Statistical applications are presented, in particular we prove a strong law of large numbers for U -statistics indexed by a one-dimensional random walk using a result of [1].Mathematics Subject Classification. 60F05, 60J15.

We consider two versions of a simple evolutionary algorithm (EA) model for protein folding at zero temperature, namely the (1 + 1)-EA on the LeadingOnes problem. In this schematic model, the structure of the protein, which is encoded as a bit-string of length n, is evolved to its native conformation through a stochastic pathway of sequential contact bindings. We study the asymptotic behavior of the hitting time, in the mean case scenario, under two different mutations: the one-flip, which flips a unique bit chosen uniformly at random in the bit-string, and the Bernoulli-flip, which flips each bit in the bit-string independently with probability c/n, for some c ∈ ℝ+ (0 ≤ c ≤ n). For each algorithm, we prove a law of large numbers, a central limit theorem, and compare the performance of the two models.