Level crossings and turning points of random hyperbolic polynomials

K. Farahmand and P. Hannigan

International Journal of Mathematics and Mathematical Sciences, 1999, vol. 22, 1-8

Abstract:

In this paper, we show that the asymptotic estimate for the expected number of K -level crossings of a random hyperbolic polynomial a 1 sinh x + a 2 sinh 2 x + ⋯ + a n sinh n x , where a j ( j = 1 , 2 , … , n ) are independent normally distributed random variables with mean zero and variance one, is ( 1 / π ) log n . This result is true for all K independent of x , provided K ≡ K n = O ( n ) . It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomial a 1 cosh x + a 2 cosh 2 x + ⋯ + a n cosh n x , with a j ( j = 1 , 2 , … , n ) as before, is also ( 1 / π ) log n .

Date: 1999
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:957474

DOI: 10.1155/S0161171299225793

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