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

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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]**H. Cramer and M. R. Leadbetter,*Stationary and related stochastic processes*, Wiley, New York, 1967. MR**0217860 (36:949)****[2]**J. E. A. Dunnage,*The number of real zeros of a random trigonometric polynomial*, Proc. London Math. Soc.**16**(1966), 53-84. MR**0192532 (33:757)****[3]**K. Farahmand,*On the number of real zeros of a random trigonometric polynomial: coefficient with non-zero mean*, Stochastic Anal. and Appl.**5**(1987), 379-386. MR**912864 (89h:42003)****[4]**-,*On the average number of level crossings of a random trigonometric polynomial*Annal. of Prob.**18**(1990). MR**1062074 (91i:60140)****[5]**S. O. Rice,*Mathematical theory of random noise*, Bell System Tech. J.**25**(1945), 46-156. MR**0011918 (6:233i)****[6]**W. Rudin,*Real and complex analysis*, 2nd ed. McGraw-Hill, 1974. MR**0344043 (49:8783)****[7]**M. Sambandham and N. Renganathan,*On the number of real zeros of a random trigonometric polynomial: coefficient with non-zero means*, J. Indian Math. Soc.**45**(1981), 193-203. MR**828871 (87g:42003)****[8]**E. C. Titchmarsh,*The theory of functions*, 2nd ed. Oxford University Press, 1939. MR**0197687 (33:5850)**

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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