We study the asymptotic behavior of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments. For the proof of tail asymptotics and integral limit theorems, we use a strong approximation of random walks by Brownian motion. For the proof of local limit theorems, we suggest a rather simple approach, which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks. We also discuss some possible applications of our results to ordered random walks and lattice path enumeration.Comment: Published at http://dx.doi.org/10.1214/13-AOP867 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Abstract. Let S 0 = 0, {Sn, n ≥ 1} be a random walk generated by a sequence of i.i.d. random variables X 1 , X 2 , ... and let τ − = min{n ≥ 1 : Sn ≤ 0} and τ + = min{n ≥ 1 : Sn > 0}. Assuming that the distribution of X 1 belongs to the domain of attraction of an α-stable law we study the asymptotic behavior, as n → ∞, of the local probabilities P(τ ± = n) and the conditional local probabilities P(Sn ∈ [x, x + ∆)|τ − > n) for fixed ∆ and x = x(n) ∈ (0, ∞) .

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We consider tightness for families of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. The model is introduced in order to mimic level lines of 2 + 1 discrete Solid-On-Solid random interfaces above a hard wall. arXiv:1809.03209v1 [math.PR]

In a recent paper of Eichelsbacher and König (2008) the model of ordered random walks has been considered. There it has been shown that, under certain moment conditions, one can construct a k-dimensional random walk conditioned to stay in a strict order at all times. Moreover, they have shown that the rescaled random walk converges to the Dyson Brownian motion. In the present paper we find the optimal moment assumptions for the construction proposed by Eichelsbacher and König, and generalise the limit theorem for this conditional process.