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Level crossings of a random trigonometric polynomial


Author: Kambiz Farahmand
Journal: Proc. Amer. Math. Soc. 111 (1991), 551-557
MSC: Primary 60G99; Secondary 42A61
MathSciNet review: 1015677
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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 $.


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  • [1] 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
  • [2] J. E. A. Dunnage, The number of real zeros of a random trigonometric polynomial, Proc. London Math. Soc. (3) 16 (1966), 53–84. MR 0192532
  • [3] 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, 10.1080/07362998708809125
  • [4] Kambiz Farahmand, On the average number of level crossings of a random trigonometric polynomial, Ann. Probab. 18 (1990), no. 3, 1403–1409. MR 1062074
  • [5] S. O. Rice, Mathematical analysis of random noise, Bell System Tech. J. 24 (1945), 46–156. MR 0011918
  • [6] Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR 0344043
  • [7] 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
  • [8] E. C. Titchmarsh, Han-shu lun, Translated from the English by Wu Chin, Science Press, Peking, 1964 (Chinese). MR 0197687

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1015677-4
Keywords: Number of real roots, Kac-Rice formula, random trigonometric polynomial
Article copyright: © Copyright 1991 American Mathematical Society