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, ∞) .

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.

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For 0 < α ≤ 2, a super-α-stable motion X in R d with branching of index 1 + β ∈ (1, 2) is considered. Fix arbitrary t > 0. If d < α/β, a dichotomy for the density function of the measure Xt holds: the density function is locally Hölder continuous if d = 1 and α > 1 + β but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local Hölder index.

We prove tightness and limiting Brownian-Gibbs description for line ensembles of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. Statistical properties of the resulting ensemble are very different from that of non-colliding Brownian bridges without self-potentials. The model itself was introduced in order to mimic level lines of 2 + 1 discrete Solid-On-Solid random interfaces above a hard wall. arXiv:1906.06533v1 [math.PR]

There is a well-known sequence of constants c n describing the growth of supercritical Galton-Watson processes Z n . With "lower deviation probabilities" we refer to P(Z n = k n ) with k n = o(c n ) as n increases. We give a detailed picture of the asymptotic behavior of such lower deviation probabilities. This complements and corrects results known from the literature concerning special cases. Knowledge on lower deviation probabilities is needed to describe large deviations of the ratio Z n+1 /Z n . The latter are important in statistical inference to estimate the offspring mean. For our proofs, we adapt the well-known Cramér method for proving large deviations of sums of independent variables to our needs.