INVARIANCE PRINCIPLES FOR LOCAL TIMES AT THE MAXIMUM OF RANDOM WALKS AND LEVY PROCESSES

• 成果类型: Article
• 署名作者: Chaumont, L.; Doney, R. A.
• 署名单位: Universite d'Angers; University of Manchester
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP512
• 发表日期: 2010
• 页码: 1368-1389
• 关键词: path decompositions; LIMIT-THEOREMS

摘要

We prove that when a sequence of Levy processes X((n)) or a normed sequence of random walks S((n)) converges as on the Skorokhod space toward a. Levy process X. the sequence L((n)) of local times at the supremum of X((n)) converges uniformly on compact sets in probability toward the local time at the supremum of X A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and descending) converges jointly in law toward the ladder processes of X As an application. we show that in general, the sequence S((n)) conditioned to stay positive converges weakly, Jointly with its local time at the future minimum, toward the corresponding functional for the limiting process X From this we deduce an invariance principle for the meander which extends known results for the case of attraction to a stable law