Level Crossings of a Random Polynomial with Hyperbolic Elements

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...

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

On Expected Number of Level Crossings of a Random Hyperbolic Polynomial

Let g1(ω), g2(ω), . . . , gn(ω) be independent and normally distributed random variables with mean zero and variance one. We show that, for large values of n, the expected number of times the random hyperbolic polynomial y = g1(ω) coshx + g2(ω) cosh 2x + · · · + gn(ω) coshnx crosses the line y = L, where L is a real number, is 1 π logn+ O(1) if L = o( √ n) or L/ √ n = O(1), but decreases steadi...

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...

Research Article On Zeros of Self-Reciprocal Random Algebraic Polynomials

This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial TN (θ)= ∑N−1 j=0 {αN− j cos( j +1/2)θ + βN− j sin( j +1/2)θ}, where αj and βj , j = 0,1,2, . . . ,N − 1, are sequences of independent identically distributed normal standard random variables. This type of random polynomial is produced in the study of random algebraic polynomials w...