First passage time for some stationary processes

For a 1-dependent stationary sequence {Xn} we first show that if u satisfies p1=p1(u)=P(X1>u)⩽0.025 and n>3 is such that 88np13⩽1, thenP{max(X1,…,Xn)⩽u}=ν·μn+O{p13(88n(1+124np13)+561)}, n>3,where ν=1−p2+2p3−3p4+p12+6p22−6p1p2,μ=(1+p1−p2+p3−p4+2p12+3p22−5p1p2)−1withpk=pk(u)=P{min(X1,…,Xk)>u}, k⩾1and|O(x)|⩽|x|.From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Ys}, a similar, in terms of P{max0⩽s⩽kTYs<u}, k=1,2 formula for P{max0⩽s⩽tYs⩽u}, t>3T and apply this formula to the process Ys=W(s+1)−W(s), s⩾0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.