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First passage time for some stationary processes

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  • Haiman, George

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.

Suggested Citation

  • Haiman, George, 1999. "First passage time for some stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 231-248, April.
    • Handle: RePEc:eee:spapps:v:80:y:1999:i:2:p:231-248
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    Citations

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    Cited by:

    1. Alexandru Amărioarei & Cristian Preda, 2015. "Approximation for the Distribution of Three-dimensional Discrete Scan Statistic," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 565-578, September.
    2. Jack Noonan & Anatoly Zhigljavsky, 2021. "Approximations for the Boundary Crossing Probabilities of Moving Sums of Random Variables," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 873-892, September.
    3. George Haiman, 2012. "1-Dependent Stationary Sequences for Some Given Joint Distributions of Two Consecutive Random Variables," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 445-458, September.
    4. Haiman, George & Preda, Cristian, 2013. "One dimensional scan statistics generated by some dependent stationary sequences," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1457-1463.
    5. Qianzhu Wu & Joseph Glaz, 2019. "Robust Scan Statistics for Detecting a Local Change in Population Mean for Normal Data," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 295-314, March.
    6. George Haiman & Cristian Preda, 2010. "A New Method of Approximating the Probability of Matching Common Words in Multiple Random Sequences," Methodology and Computing in Applied Probability, Springer, vol. 12(4), pages 775-795, December.
    7. Qianzhu Wu & Joseph Glaz, 2021. "Scan Statistics for Normal Data with Outliers," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 429-458, March.
    8. Joseph Glaz & Joseph Naus & Xiao Wang, 2012. "Approximations and Inequalities for Moving Sums," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 597-616, September.
    9. G. Haiman & C. Preda, 2006. "Estimation for the Distribution of Two-dimensional Discrete Scan Statistics," Methodology and Computing in Applied Probability, Springer, vol. 8(3), pages 373-382, September.
    10. G. Haiman & C. Preda, 2002. "A New Method for Estimating the Distribution of Scan Statistics for a Two-Dimensional Poisson Process," Methodology and Computing in Applied Probability, Springer, vol. 4(4), pages 393-407, December.

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