Level crossings of a random trigonometric polynomial

Author:
Kambiz Farahmand

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper provides an asymptotic estimate for the expected number of -level crossings of the random trigonometric polynomial , where are independent normally distributed random variables with mean and variance one. It is shown that the result for remains valid for any finite constant and any such that as .

**[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**, https://doi.org/10.1112/plms/s3-16.1.53**[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**, https://doi.org/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**, https://doi.org/10.1002/j.1538-7305.1945.tb00453.x**[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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
60G99,
42A61

Retrieve articles in all journals with MSC: 60G99, 42A61

Additional Information

DOI:
https://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