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Rough Limit Results for Level-Crossing Probabilities

Harri Nyrhinen
Journal of Applied Probability
Vol. 31, No. 2 (Jun., 1994), pp. 373-382
DOI: 10.2307/3215030
Stable URL: http://www.jstor.org/stable/3215030
Page Count: 10
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Rough Limit Results for Level-Crossing Probabilities
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Abstract

Let Y1, Y2, ⋯ be a stochastic process and M a positive real number. Define $T_{M} = \inf\{n \mid Y_{n} > M\} \(T_{M} = + \infty if Y_{n} \leqq M for n = 1, 2, \cdots\)$. We are interested in the probabilities $P(T_{M} < \infty)$ and in particular in the case when these tend to zero exponentially fast when M tends to infinity. The techniques of large deviations theory are used to obtain conditions for this and to find out the rate of convergence. The main hypotheses required are given in terms of the generating functions associated with the process (Yn).

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