Level crossings of a random trigonometric polynomial
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- by Kambiz Farahmand PDF
- Proc. Amer. Math. Soc. 111 (1991), 551-557 Request permission
Abstract:
This paper provides an asymptotic estimate for the expected number of $K$-level crossings of the random trigonometric polynomial ${g_1}\cos x + {g_2}\cos 2x + \ldots + {g_n}\cos nx$, where ${g_j}(j = 1,2, \ldots ,n)$ are independent normally distributed random variables with mean $\mu$ and variance one. It is shown that the result for $K = \mu = 0$ remains valid for any finite constant $\mu$ and any $K$ such that $({K^2}/n) \to 0$ as $n \to \infty$.References
- Harald Cramér and M. R. Leadbetter, Stationary and related stochastic processes. Sample function properties and their applications, John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0217860
- J. E. A. Dunnage, The number of real zeros of a random trigonometric polynomial, Proc. London Math. Soc. (3) 16 (1966), 53–84. MR 192532, DOI 10.1112/plms/s3-16.1.53
- K. Farahmand, On the number of real zeros of a random trigonometric polynomial: coefficients with nonzero infinite mean, Stochastic Anal. Appl. 5 (1987), no. 4, 379–386. MR 912864, DOI 10.1080/07362998708809125
- Kambiz Farahmand, On the average number of level crossings of a random trigonometric polynomial, Ann. Probab. 18 (1990), no. 3, 1403–1409. MR 1062074
- S. O. Rice, Mathematical analysis of random noise, Bell System Tech. J. 24 (1945), 46–156. MR 11918, DOI 10.1002/j.1538-7305.1945.tb00453.x
- Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0344043
- M. Sambandham and N. Renganathan, On the number of real zeros of a random trigonometric polynomial: coefficients with nonzero mean, J. Indian Math. Soc. (N.S.) 45 (1981), no. 1-4, 193–203 (1984). MR 828871
- E. C. Titchmarsh, Han-shu lun, Science Press, Peking, 1964 (Chinese). Translated from the English by Wu Chin. MR 0197687
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 111 (1991), 551-557
- MSC: Primary 60G99; Secondary 42A61
- DOI: https://doi.org/10.1090/S0002-9939-1991-1015677-4
- MathSciNet review: 1015677