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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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