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Rough limit results for level-crossing probabilities. (English) Zbl 0808.60042
Let $\{Y\sb n\}$ be a stochastic sequence. Define the level-crossing time $T = T\sb M$ by $$T = \cases \text{Inf}\{n \mid Y\sb n > M\},\quad M>0, \\ \infty \text{ if } Y\sb n \leq M \text{ for all $n \geq 1$}.\endcases$$ Using techniques of large deviations theory the author obtains $\limsup\sb{M \to \infty} M\sp{-1} \log P(T < \infty)$ and $\liminf\sb{M \to \infty} M\sp{-1} \log P(T < \infty)$ under certain regularity conditions and thereby he obtains exponential upper and lower bounds of $P(T < \infty)$ 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 {\it A. Martin-Löf} [in: Probability and mathematical statistics, Essays in Hon. of C.-G. Esseen, 129-139 (1983; Zbl 0518.62085)], {\it T. Lehtonen} and {\it 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.

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