Level crossings and turning points of random hyperbolic polynomials
In this paper, we show that the asymptotic estimate for the expected number of K-level crossings of a random hyperbolic polynomial a1sinhx+a2sinh2x+⋯+ansinhnx, where aj(j=1,2,…,n) are independent normally distributed random variables with mean zero and variance one, is (1/π)logn. This result is true...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
1999-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171299225793 |
| Summary: | In this paper, we show that the asymptotic estimate for the expected number of K-level crossings of a random hyperbolic polynomial a1sinhx+a2sinh2x+⋯+ansinhnx, where aj(j=1,2,…,n) are independent normally distributed random variables with mean zero and variance one, is (1/π)logn. This result is true for all K independent of x, provided K≡Kn=O(n). It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomial a1coshx+a2cosh2x+⋯+ancoshnx, with aj(j=1,2,…,n) as before, is also (1/π)logn. |
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| ISSN: | 0161-1712 1687-0425 |