First passage time for some stationary processes

For a 1-dependent stationary sequence {X-n} we first show that if u satisfies p(1) = p(1)(u)= P(X-1 > u) less than or equal to 0.025 and n > 3 is such that 88n p(1)(3) less than or equal to 1, then P{max(X-1, ..., X-n) less than or equal to u} = nu.mu(n) + O{p(1)(3)(88n(1 + 124np(1)(3)) + 561)}, n > 3, where n = 1 - p(2) + 2p(3) - 3p(4) + p(1)(2) + 6p(2)(2) - 6p(1)p(2), mu = (1 + p(1) - p(2) + p(3) - p(4) + 2p(1)(2) + 3p(2)(2) - 5p(1)p(2))(-1) with p(k) = p(k)(u) = P{min(X-1, ..., X-k) > u}, k greater than or equal to 1 and \O(x)\ less than or equal to \x\. From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Y-s}, a similar, in terms of P{max(0 less than or equal to s less than or equal to kT) Y-s < u}, k = 1, 2 formula for P{max(0 less than or equal to s less than or equal to t)Y(s)less than or equal to u}, t > 3T and apply this formula to the process Y-s = W(s + 1) - W(s), s greater than or equal to 0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities. (C) 1999 Published by Elsevier Science B.V. All rights reserved.