Abstract: | 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 88np131, thenP{max( X1,…, Xn) u}=ν·μ n+O{ p13(88 n(1+124 np13)+561)}, n>3, where ν=1?p2+2p3?3p4+p12+6p22?6p1p2,μ=(1+p1?p2+p3?p4+2p12+3p22?5p1p2)?1 withpk= pk( u)= P{min( X1,…, Xk)> u}, k1 and|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{max0skTYs<u}, k=1,2 formula for P{max0stYsu}, t>3T and apply this formula to the process Ys=W(s+1)?W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities. |