Conditioned random walks and Levy processes

Let X-1, X-2, ... be independent, identically distributed, zero mean random variables with (-alpha)regularly varying tails, alpha > 1. For S-n = Sigma(i=1)(n) X-i, it is known that under these distributional assumptions, P(S-n > x) similar to nP(X-1 > x) as x -> infinity, uniformly for x >= cn for any constant c > 0. Here, we show that the process M-n = max{S-i - i(mu) : i <= n}, for any constant mu >= 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-(1) S-[nt], 0 <= t <= 1 vertical bar S-n > na) and (n(-1) S-[nt], 0 <= t <= 1 vertical bar 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 Levy processes, extending the work of Konstantopoulos and Richardson ['Conditional limit theorems for spectrally positive Levy processes', Adv. in Appl. Probab. 34 (2002) 158-178] who dealt with the special case of spectrally positive processes.