Upper bounds for the maximum of a random walk with negative drift

• Autores: Johannes Kugler, Vitali Wachtel
• Localización: Journal of Applied Probability, ISSN-e 0021-9002, Vol. 50, Nº. 4, 2013, págs. 1131-1146
• Idioma: inglés
• DOI: 10.1017/s002190020001384x
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• Resumen

  • Consider a random walk Sn =?i=0n Xi negative drift. This paper deals with upper bounds for the maximum M = maxn=1 Sn of this random walk in different settings of power moment existences. As is usual for deriving upper bounds, we truncate summands. Therefore, we use an approach of splitting the time axis by stopping times into intervals of random but finite length and then choose a level of truncation on each interval. Hereby, we can reduce the problem of finding upper bounds for M to the problem of finding upper bounds for Mt = maxn=t Sn. In addition we test our inequalities in the heavy traffic regime in the case of regularly varying tails.