Reflections on Rice's formulae for level crossings - History, extensions and use

Australian and New Zealand Journal of Statistics

Volumn 46, Issue 1, 2004, Pages 173-180

  Leadbetter, M R ^a,c   Spaniolo, G V ^b  

^a UNIVERSITY OF NORTH CAROLINA   (United States)

^b GENENTECH INC   (United States)

^c The University of North Carolina at Chapel Hill   (United States)

Author keywords

Level crossings; Rice formulae; Stationary processes; Upcrossing intensities

Indexed keywords

EID: 1842815854     PISSN: 13691473     EISSN: None     Source Type: Journal    
DOI: 10.1111/j.1467-842X.2004.00322.x     Document Type: Review

References (24)

1. ADLER, R. & SAMORODNITSKY, G. (1997). Level crossings of absolutely continuous stationary symmetric α-stable processes. Ann. Appl. Probab. 7, 460-493.
2. ADLER, R., SAMORODNITSKY, G. & GADRJCH, T. (1993). The expected number of level crossings for stationary, harmonizable, symmetric, stable processes. Ann. Appl. Probab. 3, 553-575.
3. ALBIN, J.M.P. (2000). Extremes and upcrossing intensities for P-differentiable stationary processes. Stoch. Proc. Appl. 87, 199-234.
4. ALBIN, J.M.P. & LEADBETTER, M.R. (1999). Asymptotic behavior of conditional laws and moments of α-stable random vectors, with application to upcrossing intensities. Ann. Probab. 27, 1468-1500.
5. BRILLINGER, D.R. (1972). On the number of solutions of systems of random equations. Ann. Math. Statist. 43, 534-540.
6. BULINSKAYA, E.V. (1961). On the mean number of crossings of a level by a stationary Gaussian process. Theory Probab. Appl. 6, 435-438.
7. DOOB, J.L. (1953). Stochastic Processes. New York: Wiley.
8. GEMAN, D. & HOROWITZ, J. (1973). Occupation times for smooth stationary processes. Ann. Probab. 1, 131-137.
9. ITO, K. (1964). The expected number of zeros of continuous stationary Gaussian processes. J. Math. Kyoto Univ. 3, 207-216.
10. IVANOV, V.A. (1960). On the average number of crossings of a level by sample functions of a stochastic process. Theory Probab. Appl. 5, 319-323.
11. KAC, M. (1943). On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc. 49, 314-320.
12. LEADBETTER, M.R. (1966). On crossings of levels and curves by a wide class of stochastic processes. Ann. Math. Staust. 37, 260-267.
13. LEADBETTER, M.R. & SPANIOLO, G.V. (2002). Marked Crossings by Stationary Processes and Their Applications. UNC Stat. Dept. Stoch. Proc. Ctr. TR# 526.
14. LEADBETTER, M.R., LINDGREN, G. & ROOTZÉN, H. (1983). Extremes and Related Properties of Random Sequences and Processes. New York: Springer Verlag.
15. MARCUS, M.B. (1977). Level crossings of a stochastic process with absolutely continuous sample paths. Ann. Probab. 5, 52-71.
16. MARCUS, M.B. (1989). Some bounds for the expected number of level crossings of symmetric harmonizable p-stable processes. Stochastic Process. Appl. 33, 217-231.
17. MICHNA, Z. & RYCHLIK, I. (1995). Expected number of level crossings for certain symmetric α-stable processes. Comm. Statist. Stochastic Models 11, 1-19.
18. RICE, S.O. (1944). Mathematical analysis of random noise. Bell Syst. Tech. J. 23, 292-332.
19. RICE, S.O. (1945). Mathematical analysis of random noise. Bell Syst. Tech. J. 24, 46-156.
20. RYCHLIK, I. (2000). On some reliability applications of Rice's Formula for the intensity of level crossings. Extremes 3, 331-348.
21. SAMORODNITSKY, G. & TAQQU, M.S. (1994). Stable Non-Gaussian Random Processes. London: Chapman and Hall.
22. SPANIOLO, G.V. (2000). Rice's Formula and Palm Probabilities with Applications to Structural Reliability. PhD Dissertation, University of North Carolina.
23. YLVISAKER, N.D. (1965). The expected number of zeros of a stationary Gaussian process. Ann. Math. Statist. 36, 1043-1046.
24. ZÄHLE, U. (1984). A general Rice formula, Palm measures, and horizontal-window conditioning for random fields. Stochastic Process. Appl. 17, 265-283.