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Rough limit results for level-crossing probabilities. (English) Zbl 0808.60042

Let {Y n } be a stochastic sequence. Define the level-crossing time T=T M by

T=Inf{nY n >M},M>0,ifY n Mforalln1·

Using techniques of large deviations theory the author obtains lim sup M M -1 logP(T<) and lim inf M M -1 logP(T<) under certain regularity conditions and thereby he obtains exponential upper and lower bounds of P(T<) for large M. As examples his regularity conditions are satisfied by sums of i.i.d. sequences, Markov additive processes and partial sums of moving averages. His results extend and complement those of A. Martin-Löf [in: Probability and mathematical statistics, Essays in Hon. of C.-G. Esseen, 129-139 (1983; Zbl 0518.62085)], T. Lehtonen and H. Nyrhinen [Adv. Appl. Probab. 24, No. 4, 858-874 (1992; Zbl 0779.65003) and Scand. Actuarial J. 1992, No. 1, 60-75 (1992; Zbl 0755.62080)] and of many others.

MSC:
60G40Stopping times; optimal stopping problems; gambling theory
60F10Large deviations
62P05Applications of statistics to actuarial sciences and financial mathematics