## Saddlepoint approximations and nonlinear boundary crossing probabilities of Markov random walks.(English)Zbl 1029.60058

The authors present saddlepoint approximations for Markov random walks $$S_n$$ which are used to evaluate the probability that $$(j-i)g((S_j -S_i)/(j-i))$$ exceeds a threshold value for certain sets of $$(i,j)$$. The special case $$g(x)=x$$ reduces to the usual scan statistic in change-point detection problems, and many generalized likelihood ratio detection schemes are also of this form with suitably chosen $$g$$. This boudary crossing probability is used to derive both the asymptotic Gumbel-type distribution of scan statistics and the asymptotic exponential distribution of the waiting time to false alarm in sequential change-point detection. Combinig these saddlepoint approximations with truncation arguments and geometric integration theory also yields asymptotic formulas for other nonlinear boundary crossing probabilities of Markov random walks satisfying certain conditions.

### MSC:

 60J05 Discrete-time Markov processes on general state spaces 60F10 Large deviations 60G60 Random fields 60F05 Central limit and other weak theorems 60G40 Stopping times; optimal stopping problems; gambling theory
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### References:

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