Abstract
We consider the problem of finding the probability of the event that a continuous random process reaches the boundary of a region first time on a given range of the independent variable. We propose a new approach to estimating the said probability related to studying the so-called conditional probabilities of a horizontal window: a) conditional probability of the event that at the moment when component ξ 1(x) of the n-dimensional process ξ(x) = {ξ 1(x), …, ξ n (x)} first drops under a given level on interval [x, x + Δx) constraint (ξ 2, …, ξ n ) ∈ D ⊂ R n−1 holds, where D is a given region, given that it has dropped under this level at all; b) conditional probability, under the same condition as a), of the event that up until the moment of the first entry of component ξ 1(x) under a given level on interval [x, x + Δx) this component had already crossed this level a certain number of times.
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Semakov, S.L., First Arrival of a Stochastic Process at the Boundary, Autom. Remote Control, 1988, vol. 49, no. 6, part 1, pp. 757–764.
Pontryagin, L.S., Andronov, A.A., and Vitt, A.A., On Statistical Analysis of Dynamical Systems, Zh. Eksp. Teor. Fiz., 1933, vol. 3, no. 3, pp. 165–180.
Semakov, S.L., Vybrosy sluchainykh protsessov: prilozheniya v aviatsii (Outliers of Random Processes: Applications in Aviation), Moscow: Nauka, 2005.
Semakov, S.L., The Probability of the First Hitting of a Level by a Component of a Multidimensional Process on a Prescribed Interval under Restrictions on the Remaining Components, Theor. Prob. App., 1989, vol. 34. no. 2, pp. 357–361.
Kac, M. and Slepian, D., Large Excursions of Gaussian Processes, Ann. Math. Statist., 1959, vol. 30, pp. 1215–1228.
Cramer, H. and Leadbetter, M.R., Stationary and Related Stochastic Processes, New York: Wiley, 1967. Translated under the title Statsionarnye sluchainye protsessy, Moscow: Mir, 1969.
Bulinskaya, E.V., On the Average Number of Times a Stationary Gaussian Process Crosses a Certain Level, Theor. Veroyat. Primen., 1961, vol. 6, no. 4, pp. 474–478.
Belyaev, Yu.K., A General Formula for the Average Number of Intersections for Random Processes and Fields, in Vybrosy sluchainykh polei (Outliers in Random Fields), Moscow: Mosk. Gos. Univ., 1972, no. 29, pp. 38–45.
Slepian, D., The One-sided Barrier Problem for Gaussian Noise, Bell Syst. Techn. J., 1962, vol. 41, no. 2, pp. 463–501.
Azais, A.-M. and Wschebor, M., Levels Sets and Extrema of Random Processes and Fields, New Jersey: Wiley, 2009.
Rice, S.O., Mathematical Analysis of Random Noise, Bell Syst. Techn. J., 1945, vol. 24, no. 1, pp. 46–156.
Semakov, S.L., The Application of the Known Solution of a Problem of Attaining the Boundaries by Non-Markovian Process to Estimation of Probability of Safe Airplane Landing, J. Comput. Syst. Sci. Int., 1996, vol. 35, no. 2, pp. 302–308.
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Original Russian Text © S.L. Semakov, 2015, published in Avtomatika i Telemekhanika, 2015, No. 4, pp. 80–96.
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Semakov, S.L. Estimating the probability that a multidimensional random process reaches the boundary of a region. Autom Remote Control 76, 613–626 (2015). https://doi.org/10.1134/S0005117915040062
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DOI: https://doi.org/10.1134/S0005117915040062