Level crossings of a random polynomial with hyperbolic elements

Author:
K. Farahmand

Journal:
Proc. Amer. Math. Soc. **123** (1995), 1887-1892

MSC:
Primary 60H99; Secondary 42A05

DOI:
https://doi.org/10.1090/S0002-9939-1995-1264810-1

MathSciNet review:
1264810

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper provides an asymptotic estimate for the expected number of *K*-level crossings of a random hyperbolic polynomial ${g_1}\cosh x + {g_2}\cosh 2x + \cdots + {g_n}\cosh nx$, where ${g_j}(j = 1,2, \ldots ,n)$ are independent normally distributed random variables with mean zero, variance one and *K* is any constant independent of *x*. It is shown that the result for $K = 0$ remains valid as long as $K \equiv {K_n} = O(\sqrt n )$.

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Additional Information

Keywords:
Gaussian process,
number of real roots,
Kac-Rice formula,
algebraic polynomials,
trigonometric polynomials,
fixed probability space

Article copyright:
© Copyright 1995
American Mathematical Society