Level crossings and turning points of random hyperbolic 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).

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...

Bounds of the Number of Level Crossings of the Random Algebraic Polynomials

-In this paper we have estimate bounds of the number of level crossings of the random algebraic polynomials     n k k k n x t a x f 0 0 ) ( ) 1 , ( where , 1 0 , ) (    t t t ak are dependent random variables assuming real values only and following the normal distribution with mean zero and joint density function      M M s a ' ) 2 / 1 ( exp ) 2 ( / 2 / 1   . There exists an integ...