Level Crossings and Turning Points of Random Hyperbolic Polynomials

نویسندگان

  • K. FARAHMAND
  • P. HANNIGAN
چکیده

In this paper, we show that the asymptotic estimate for the expected number of K-level crossings of a random hyperbolic polynomial a1 sinhx+a2 sinh2x+···+ an sinhnx, 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 a1 coshx+a2 cosh2x +···+an coshnx, with aj(j = 1,2, . . . ,n) as before, is also (1/π) logn.

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تاریخ انتشار 1999