Том 62 ТЕОРИЯ ВЕРОЯТНОСТЕЙ И ЕЕ ПРИМЕНЕНИЯ 2017 2017 г. c MALLEIN B.∗ N-BRANCHING RANDOM WALK WITH α-STABLE SPINE1) Вып у с к 2 На вещественной прямой рассматривается система частиц с ветвлением и селекцией, введенная Э. <...> Только N самых правых потомков выживают, чтобы дать потомство на следующем шаге. <...> Гуэре [1] изучали скорость, с которой движется облако частиц, в предположении, что хвосты сдвигов убывают экспоненциально;Ж. <...> Let L be the law of a random point process on R. Brunet, Derrida et al. introduced in [4], [5] a discrete-time branching-selection particle system on R in which the size of the population is limited by some integer N. This process evolves as follows: for any n ∈ N, every individual alive at the n-th generation dies giving birth to children around its current position, according to an independent version of a point process of law L. Only the N children with the largest position are kept alive and form the (n + 1)-st generation of the process. <...> We write (xN n (1), . . . ,xN n (N)) for the positions at time n of individuals in the process, ranked in the decreasing order. <...> This process is called the N-branching random walk, or N-BRW for short. <...> Laboratoire de Probabilit´ (Paris 6); D´ es et Mod` epartement de Math´ ematiques et Applications, ´ eles Al´ eatoires, Universit´ Ecole Normale Sup´ Paris, France; e-mail: bastien.mallein@ens.fr 1)Research partially supported by the ANR project MEMEMO2. e Pierre et Marie Curie erieure, 366 In [1], B´ er´ Mallein B. erard andGou´ e proved that under some appropriate integrability conditions, the cloud of particles drifts at some deterministic speed vN := limn→+∞ xN n (1) n = limn→+∞ xN n (N) n (lnN)2 , C a.s., and obtained the following asymptotic behavior for vN: v∞ −vN ∼N→+∞ (1.1) (1.2) in which C is an explicit positive constant that depends only on the law L. Their argument is based <...>