Details

Autor(en) / Beteiligte

• Doney, R. A
• Jones, E. M

Titel

Conditioned random walks and Lévy processes

Ist Teil von

• The Bulletin of the London Mathematical Society, 2012-02, Vol.44 (1), p.139-150

Ort / Verlag

Oxford University Press

Erscheinungsjahr

2012

Link zum Volltext

Quelle

Wiley Online Library Journals Frontfile Complete

Beschreibungen/Notizen

• Let X 1, X 2, ... be independent, identically distributed, zero mean random variables with (−α)-regularly varying tails, α>1. For , it is known that under these distributional assumptions, (S n >x)∼ n (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 −1 S [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.