On a Uniform Random Walk Conditioned to Stay Positive
نویسنده
چکیده
For arbitrary N, M ∈ N we consider the ’uniform random walk’ (Sn)n≥0 with step sizes −N,−N +1, . . . , 0, , . . . , M and S0 = a ∈ N, conditioned to stay positive. The conditional generating functions of S0, S1, S2, . . . are expressed in terms of the coefficients of the expansion of the function PN+M (x) n, where Pi(x) = 1+x+· · ·+xi (i ∈ N, n ∈ Z), in ascending powers of x or, for M = 1, using alternatively the inverse function of x/PN+1(x), thereby
منابع مشابه
An Invariance Principle for Random Walk Bridges Conditioned to Stay Positive
We prove an invariance principle for the bridge of a random walk conditioned to stay positive, when the random walk is in the domain of attraction of a stable law, both in the discrete and in the absolutely continuous setting. This includes as a special case the convergence under diffusive rescaling of random walk excursions toward the normalized Brownian excursion, for zero mean, finite varian...
متن کاملUpper and Lower Space-time Envelopes for Oscillating Random Walks Conditioned to Stay Positive
We provide integral tests for functions to be upper and lower space time envelopes for random walks conditioned to stay positive. As a result we deduce a `Hartman-Winter' Law of the Iterated Logarithm for random walks conditioned to stay positive under a third moment assumption. We also show that under a second moment assumption the conditioned random walk grows faster than n 1=2 (log n) ?(1+")...
متن کاملInvariance principles for random walks conditioned to stay positive
Let {Sn} be a random walk in the domain of attraction of a stable law Y, i.e. there exists a sequence of positive real numbers (an) such that Sn/an converges in law to Y. Our main result is that the rescaled process (S⌊nt⌋/an, t ≥ 0), when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable Lévy process conditioned to stay positive. Under so...
متن کاملLaw of the iterated logarithm for oscillating random walks conditioned to stay non-negative
We show that under a 3+ δ moment condition (where δ > 0) there exists a ‘Hartman-Winter’ Law of the Iterated Logarithm for random walks conditioned to stay non-negative. We also show that under a second moment assumption the conditioned random walk eventually grows faster than n (logn) for any ε > 0 and yet slower than n (logn) . The results are proved using three key facts about conditioned ra...
متن کاملRandom Walks in Cones
We study the asymptotic behaviour 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...
متن کامل