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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: 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

DOI: https://doi.org/10.1090/S0002-9939-1995-1264810-1
Keywords: Gaussian process, number of real roots, Kac-Rice formula, algebraic polynomials, trigonometric polynomials, fixed probability space
Article copyright: © Copyright 1995 American Mathematical Society

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