First passage time for some stationary processes

George Haiman

Stochastic Processes and their Applications, 1999, vol. 80, issue 2, 231-248

Abstract: For a 1-dependent stationary sequence {Xn} we first show that if u satisfies p1=p1(u)=P(X1>u)[less-than-or-equals, slant]0.025 and n>3 is such that 88np13[less-than-or-equals, slant]1, thenP{max(X1,...,Xn)[less-than-or-equals, slant]u}=[nu]·[mu]n+O{p13(88n(1+124np13)+561)}, n>3,where [nu]=1-p2+2p3-3p4+p12+6p22-6p1p2,[mu]=(1+p1-p2+p3-p4+2p12+3p22-5p1p2)-1withpk=pk(u)=P{min(X1,...,Xk)>u}, k[greater-or-equal, slanted]1andO(x)[less-than-or-equals, slant]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[less-than-or-equals, slant]s[less-than-or-equals, slant]kTYs 3T and apply this formula to the process Ys=W(s+1)-W(s), s[greater-or-equal, slanted]0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.

Keywords: Increments; of; Wiener; process; -; first; passage; time; Stationary; T-dependent; processes (search for similar items in EconPapers)
Date: 1999
References: View complete reference list from CitEc
Citations: View citations in EconPapers (10)

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