Level crossings of a random polynomial with hyperbolic elements

Farahmand, K.

doi:10.1090/S0002-9939-1995-1264810-1

Level crossings of a random polynomial with hyperbolic elements
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Proc. Amer. Math. Soc. 123 (1995), 1887-1892 Request permission

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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