We consider the minimum of a super-critical branching random walk. Addario-Berry and Reed [Ann. Probab. 37 (2009) 1044-1079] proved the tightness of the minimum centered around its mean value. We show that a convergence in law holds, giving the analog of a well-known result of Bramson [Mem. Amer. Math. Soc. 44 (1983) iv+190] in the case of the branching Brownian motion.Comment: Published in at http://dx.doi.org/10.1214/12-AOP750 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org) Compared to the published version, we extended our result to the case of an infinite number of children. To this end, we added Corollary 3.5 and slightly changed the proof of Theorem 1.1. arXiv admin note: text overlap with arXiv:1107.2543 by other author

We consider the boundary case (in the sense of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609-631] in a one-dimensional super-critical branching random walk, and study the additive martingale (Wn). We prove that, upon the system's survival, n 1/2 Wn converges in probability, but not almost surely, to a positive limit. The limit is identified as a constant multiple of the almost sure limit, discovered by Biggins and Kyprianou [Adv. in Appl. Probab. 36 (2004) 544-581], of the derivative martingale.

Summary. We consider a transient random walk (X n ) in random environment on a Galton-Watson tree. Under fairly general assumptions, we give a sharp and explicit criterion for the asymptotic speed to be positive. As a consequence, situations with zero speed are revealed to occur. In such cases, we prove that X n is of order of magnitude n Λ , with Λ ∈ (0, 1). We also show that the linearly edge reinforced random walk on a regular tree always has a positive asymptotic speed, which improves a recent result of Collevecchio [1].Key words. Random walk in random environment, reinforced random walk, law of large numbers, Galton-Watson tree.

Summary. We give an expression of the speed of the biased random walk on a Galton-Watson tree. In the particular case of the simple random walk, we recover the result of Lyons, Pemantle and Peres [8]. The proof uses a description of the invariant distribution of the environment seen from the particle.

We show that the trace of the null recurrent biased random walk on a Galton-Watson tree properly renormalized converges to the Brownian forest. Our result extends to the setting of the random walk in random environment on a Galton-Watson tree.