On Expected Number of Level Crossings of a Random Hyperbolic Polynomial

On the average number of sharp crossings of certain Gaussian random polynomials

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 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 asym...

Level Crossings of a Random Polynomial with Hyperbolic Elements

This paper provides an asymptotic estimate for the expected number of AMevel crossings of a random hyperbolic polynomial gi cosh x + g2 cosh 2x + ■ ■ ■ + g„ cosh nx , where g¡ (j = 1, 2,..., 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 = K„ = 0{s/n).

Expected Number of Local Maxima of Some Gaussian Random Polnomials

On the Average Number of Sharp Crossings of Certain Gaussian Random Polynomials

Let Qn(x) = ∑n i=0 Aix i be a random algebraic polynomial where the coefficients A0, A1, · · · form a sequence of centered Gaussian random variables. Moreover, assume that the increments ∆j = Aj−Aj−1, j = 0, 1, 2, · · · are independent, assuming A−1 = 0. The coefficients can be considered as n consecutive observations of a Brownian motion. We obtain the asymptotic behaviour of the expected numb...