Level crossings of a random trigonometric polynomial
Author:
Kambiz Farahmand
Journal:
Proc. Amer. Math. Soc. 111 (1991), 551557
MSC:
Primary 60G99; Secondary 42A61
MathSciNet review:
1015677
Fulltext PDF Free Access
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, 1967. MR 0217860
(36 #949)
 [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 (33 #757)
 [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
(89h:42003), http://dx.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
(91i:60140)
 [5]
S.
O. Rice, Mathematical analysis of random noise, Bell System
Tech. J. 24 (1945), 46–156. MR 0011918
(6,233i)
 [6]
Walter
Rudin, Real and complex analysis, 2nd ed., McGrawHill Book
Co., New York, 1974. McGrawHill Series in Higher Mathematics. MR 0344043
(49 #8783)
 [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. 14, 193–203 (1984). MR 828871
(87g:42003)
 [8]
E.
C. Titchmarsh, Hanshu lun, Translated from the English by Wu
Chin, Science Press, Peking, 1964 (Chinese). MR 0197687
(33 #5850)
 [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), 5384. MR 0192532 (33:757)
 [3]
 K. Farahmand, On the number of real zeros of a random trigonometric polynomial: coefficient with nonzero mean, Stochastic Anal. and Appl. 5 (1987), 379386. 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), 46156. MR 0011918 (6:233i)
 [6]
 W. Rudin, Real and complex analysis, 2nd ed. McGrawHill, 1974. MR 0344043 (49:8783)
 [7]
 M. Sambandham and N. Renganathan, On the number of real zeros of a random trigonometric polynomial: coefficient with nonzero means, J. Indian Math. Soc. 45 (1981), 193203. MR 828871 (87g:42003)
 [8]
 E. C. Titchmarsh, The theory of functions, 2nd ed. Oxford University Press, 1939. MR 0197687 (33:5850)
Similar Articles
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:
http://dx.doi.org/10.1090/S00029939199110156774
PII:
S 00029939(1991)10156774
Keywords:
Number of real roots,
KacRice formula,
random trigonometric polynomial
Article copyright:
© Copyright 1991 American Mathematical Society
