Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T01:06:10.112Z Has data issue: false hasContentIssue false
  • English
  • Français

The average number of real zeros of a random trigonometric polynomial

Published online by Cambridge University Press:  24 October 2008

Minaketan Das
Minaketan Das
Affiliation:
F.M. College, Balasore, India
Get access Rights & Permissions [Opens in a new window]

Abstract

Let a1, a2,… be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity; let b1, b2,… be a set of positive constants. In this work, we obtain the average number of zeros in the interval (0, 2π) of trigonometric polynomials of the form

for large n. The case when bk = kσ (σ > − 3/2;) is studied in detail. Here the required average is (2σ + 1/2σ + 3)½.2n + o(n) for σ ≥ − ½ and of order n3/2; + σ in the remaining cases.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 3 , July 1968 , pp. 721 - 730
Copyright
Copyright © Cambridge Philosophical Society 1968

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

(1)Cramer, H.Mathematical methods of Statistics (Princeton University Press, 1958).Google Scholar
(2)Das, M. The number of real zeros of certain random trigonometric polynomials—submitted to Math. Student.Google Scholar
(3)Dunnage, J. E. A.The number of real zeros of a random trigonometric polynomial. Proc. London Math. Soc. 16 (1966), 53–84.Google Scholar
(4)Kac, M.Probability and related topics in Physical Sciences (New York, Interscience, 1959).Google Scholar
(5)Titchmarsh, E. C.Theory of Functions (Oxford, 1947).Google Scholar