Saddlepoint approximations are developed for Markov random walks $S_n$
and 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 $g$. We make use of this boundary crossing probability 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. Combining
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 minorization conditions.
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