Conditioned random walks and Lévy processes

Let X 1, X 2, ⋯ be independent, identically distributed, zero mean random variables with (-α)-regularly varying tails, α > 1. For S n = ∑ n i=1 x i, it is known that under these distributional assumptions, ℙ(S n > x)∼ nℙ(X 1 > x) as x → ∞, uniformly for x ≥ cn for any constant c > 0. Here, we show that the process M n = max {S i-iμ : i ≤ n}, for any constant μ ≥ 0, behaves in a similar manner. This allows us to generalize Durrett's results ['Conditioned limit theorems for random walks with negative drift', Z. Wahrscheinlichkeitstheorie verw. Gebiete 52 (1980) 277-287], by showing that, without any further assumptions, both (n -1S [nt], 0 ≤ t ≤ 1|S n > na) and (n -1 S [nt], 0 ≤ t ≤ 1|M n > na) for any constant a > 0 converge weakly to a simple process consisting of a single 'large jump'.We show that similar results hold for general Lévy processes, extending the work of Konstantopoulos and Richardson ['Conditional limit theorems for spectrally positive Lévy processes', Adv. in Appl. Probab. 34 (2002) 158-178] who dealt with the special case of spectrally positive processes. © 2011 London Mathematical Society.