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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: This paper provides an asymptotic estimate for the expected number of $ K$-level crossings of the random trigonometric polynomial $ {g_1}\cos x + {g_2}\cos 2x + \ldots + {g_n}\cos nx$, where $ {g_j}(j = 1,2, \ldots ,n)$ are independent normally distributed random variables with mean $ \mu $ and variance one. It is shown that the result for $ K = \mu = 0$ remains valid for any finite constant $ \mu $ and any $ K$ such that $ ({K^2}/n) \to 0$ as $ n \to \infty $.


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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

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