Asymptotic theory for curve-crossing analysis

Stochastic Processes and their Applications

Volumn 117, Issue 7, 2007, Pages 862-877

  Zhao, Zhibiao ^a   Wu, Wei Biao ^a  

^a UNIVERSITY OF CHICAGO   (United States)

Author keywords

Central limit theorem; Curve crossing; Linear processes; Multiple Wiener It integral; Non central limit theorem; Nonlinear time series

Indexed keywords

INTEGRAL EQUATIONS; LINEAR SYSTEMS; NONLINEAR ANALYSIS; THEOREM PROVING; TIME SERIES ANALYSIS;

ASYMPTOTIC DISTRIBUTION; ASYMPTOTIC THEORY; LINEAR PROCESSES; NONLINEAR TIME SERIES;

ASYMPTOTIC ANALYSIS;

EID: 34248585381     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2006.10.010     Document Type: Article

References (46)

1. Benzaquen S., and Cabaña E.M. The expected measure of the level sets of a regular stationary Gaussian process. Pacific J. Math. 103 (1982) 9-16
2. Bulinskaya E.V. On the mean number of crossings of a level by a stationary Gaussian process. Theory Probab. Appl. 6 (1961) 435-438
3. Chow Y.S., and Teicher H. Probability Theory (1998), Springer-Verlag, New York
4. Cuzick J. A central limit theorem for the number of zeros of a stationary Gaussian process. Ann. Probab. 4 (1976) 547-556
5. Dedecker J., and Prieur C. New dependence coefficients. Examples and applications to statistics. Probab. Theory Related Fields 132 (2005) 203-236
6. De La Peña V., and Giné E. Decoupling: From dependence to independence: Randomly stopped processes. U-statistics and Processes, Martingales and Beyond (1999), Springer-Verlag, New York
7. Diaconis P., and Freedman D. Iterated random functions. SIAM Rev. 41 (1999) 41-76
8. Duflo M. Random Iterative Models (1997), Springer-Verlag, Heidelberg, Germany
9. Engle R.F. Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation. Econometrica 50 (1982) 987-1007
10. Fan J., and Yao Q. Nonlinear Time Series: Nonparametric and Parametric Methods (2003), Springer, New York
11. B. Gajic, K.K. Paliwal, Robust speech recognition using features based on zero crossings with peak amplitudes, in: Proc. IEEE Intern. Conf. on Acoustics, Speech and Signal Processing, vol. I, Hong Kong, April 2003, pp. 64-67
12. Giraitis L., Koul H.L., and Surgailis D. Asymptotic normality of regression estimators with long memory errors. Statist. Probab. Lett. 29 (1996) 317-335
13. Haggan V., and Ozaki T. Modelling nonlinear random vibrations using an amplitude-dependent autoregressive time series model. Biometrika 68 (1981) 189-196
14. Ho H.-C., and Hsing T. On the asymptotic expansion of the empirical process of long-memory moving averages. Ann. Statist. 24 (1996) 992-1024
15. Ho H.-C., and Hsing T. A decomposition for generalized U-statistics of long-memory processes. In: Doukhan P., Oppenheim G., and Taqqu M.S. (Eds). Theory and Applications of Long-range Dependence (2003), Birkhäuser, Boston 143-155
16. Hoeffding W., and Robbins H. The central limit theorem for dependent random variables. Duke Math. J. 15 (1948) 773-780
17. Kallianpur G. Some ramifications of Wieners ideas on nonlinear prediction. Norbert Wiener, Collected Works with Commentaries (1981), MIT Press, MA 402-424
18. Kasahara Y., and Maejima M. Weighted sums of i.i.d. random variables attracted to integrals of stable processes. Probab. Theory Related Fields 78 (1988) 75-96
19. Kedem B. On the sinusoidal limit of stationary time series. Ann. Statist. 12 (1984) 665-674
20. Kedem B. Higher-order crossings in time series model identification. Technometrics 29 (1987) 193-204
21. Kedem B. Time Series Analysis by Higher Order Crossings (1994), IEEE Press, New York
22. Kratz M.F., and León J.R. Hermite polynomial expansion for non-smooth functionals of stationary Gaussian processes: Crossings and extremes. Stochastic Process Appl. 66 (1997) 237-252
23. M.F. Kratz, J.R. León, Curve crossings and specular points, d'après Longuet-Higgins, 2005, preprint
24. Kuipers L., and Niederreiter H. Uniform Distribution of Sequences (1974), Wiley, New York
25. Letac G. A contraction principle for certain Markov chains and its applications. Random Matrices and Their Applications. Contemp. Math. vol. 50 (1986), Amer. Math. Soc., Providence, RI 263-273
26. Longuet-Higgins M.S. Reflection and refraction at a random moving surface, I, II, III. J. Opt. Soc. Amer. 50 (1960) 838-856
27. Major P. Multiple Wiener-Itô Integrals: With Applications to Limit Theorems (1981), Springer, New York
28. Mokkadem A. Propriétés de mélange des processus autorǵressifs polynomiaux. Ann. Inst. H. Poincaré Probab. Statist. 26 (1990) 219-260
29. Nicholls D.F., and Quinn B.G. Random Coefficient Autoregressive Models: An Introduction (1982), Springer-Verlag, New York
30. Niederjohn R.J., and Castelaz P.F. Zero-crossing analysis methods for speech recognition. Proc. 1978 Conf. on Pattern Recognition and Image Processing vol. CH1318 (1978), IEEE, New York 507-513
31. Rice S.O. Mathematical analysis of random noise. Bell Syst. Tech. J. 24 (1945) 46-156
32. Romano J.P., and Wolf M. A more general central limit theorem for m-dependent random variables with unbounded m. Statist. Probab. Lett. 47 (2000) 115-124
33. Rosenblatt M. Stationary processes as shifts of functions of independent random variables. J. Math. Mech. 8 (1959) 665-681
34. Sánchez de Naranjo M.V. A central limit theorem for non-linear functionals of stationary Gaussian vector processes. Statist. Probab. Lett. 22 (1995) 223-230
35. J. Saunders, Real-time discrimination of broadcast speech/music, in: Proc. ICASSP'96, vol. II, Atlanta, GA, 1996, pp. 993-996
36. Shimizu K., and Tanaka M. Expected number of level-crossings for a strictly stationary ellipsoidal process. Statist. Probab. Lett. 64 (2003) 305-310
37. Slud E.V. MWI representation of the number of curve-crossings by a differentiable Gaussian process, with applications. Ann. Probab. 22 (1994) 1355-1380
38. Surgailis D. Zones of attraction of self-similar multiple integrals. Lithuanian Math. J. 22 (1982) 327-340
39. Surgailis D. Non-CLTs: U-statistics, multinomial formula and approximations of multiple Itô-Wiener integrals. In: Doukhan P., Oppenheim G., and Taqqu M.S. (Eds). Theory and Applications of Long-range Dependence (2003), Birkhäuser, Boston 129-142
40. Tong H. Nonlinear Time Series Analysis: A Dynamic Approach (1990), Oxford University Press, Oxford
41. Wu W.B. Central limit theorems for functionals of linear processes and their applications. Statist. Sinica 12 (2002) 635-649
42. Wu W.B. Additive functionals of infinite-variance moving averages. Statist. Sinica 13 (2003) 1259-1267
43. Wu W.B. Empirical processes of long-memory sequences. Bernoulli 9 (2003) 809-831
44. Wu W.B., and Min W. On linear processes with dependent innovations. Stochastic Process Appl. 115 (2005) 939-958
45. Wu W.B., and Shao X. Limit theorems for iterated random functions. J. Appl. Probab. 41 (2004) 425-436
46. Ylvisaker N.D. The expected number of zeros of a stationary Gaussian process. Ann. Math. Statist. 36 (1965) 1043-1046