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)[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<u}, k=1,2 formula for P{max0[less-than-or-equals, slant]s[less-than-or-equals, slant]tYs[less-than-or-equals, slant]u}, t>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.
Year of publication: |
1999
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Authors: | Haiman, George |
Published in: |
Stochastic Processes and their Applications. - Elsevier, ISSN 0304-4149. - Vol. 80.1999, 2, p. 231-248
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Publisher: |
Elsevier |
Subject: | Increments of Wiener process | first passage time Stationary T-dependent processes |
Saved in:
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