Doney, R; Jones, E; (2012) Large deviation results for random walks conditioned to stay positive. Electronic Communications in Probability , 17 (38) pp. 1-11. 10.1214/ecp.v17-2282.

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Abstract

Let X1, X2, ... denote independent, identically distributed random variables with common distribution F, and S the corresponding random walk with ρ := limn→∞ P(Sn > 0) and τ := inf{n ≥ 1 : Sn ≤ 0}. We assume that X is in the domain of attraction of an α-stable law, and that P(X ∈ [x, x + ∆)) is regularly varying at infinity, for fixed ∆ > 0. Under these conditions, we find an estimate for P(Sn ∈ [x, x + ∆)|τ > n), which holds uniformly as x/cn → ∞, for a specified norming sequence cn. This result is of particular interest as it is related to the bivariate ladder height process ((Tn, Hn), n ≥ 0), where Tr is the rth strict increasing ladder time, and Hr = STr the corresponding ladder height. The bivariate renewal mass function g(n, dx) = P∞ r=0 P(Tr = n, Hr ∈ dx) can then be written as g(n, dx) = P(Sn ∈ dx|τ > n)P(τ > n), and since the behaviour of P(τ > n) is known for asymptotically stable random walks, our results can be rephrased as large deviation estimates of g(n, [x, x + ∆)).

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