scholarly journals Level crossings and turning points of random hyperbolic polynomials

1999 ◽  
Vol 22 (3) ◽  
pp. 579-586
Author(s):  
K. Farahmand ◽  
P. Hannigan
Keyword(s):  
Asymptotic Estimate ◽  
Turning Points ◽  
Random Variables ◽  
Random Polynomial ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Level Crossings ◽  
Hyperbolic Polynomials ◽  
Normally Distributed

In this paper, we show that the asymptotic estimate for the expected number ofK-level crossings of a random hyperbolic polynomiala1sinhx+a2sinh2x+⋯+ansinhnx, whereaj(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 allKindependent ofx, providedK≡Kn=O(n). It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomiala1coshx+a2cosh2x+⋯+ancoshnx, withaj(j=1,2,…,n)as before, is also(1/π)logn.

scholarly journals Level crossings of a random trigonometric polynomial with dependent coefficients

1995 ◽  
Vol 58 (1) ◽  
pp. 39-46
Author(s):  
K. Farahmand
Keyword(s):  
Trigonometric Polynomial ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Expected Number ◽  
Level Crossings ◽  
Random Trigonometric Polynomial ◽  
Normally Distributed

AbstractThis paper provides an asymptotic estimate for the expected number of K-level crossings of the random trigonometric polynomial g1 cos x + g2 cos 2x+ … + gn cos nx where gj (j = 1, 2, …, n) are dependent normally distributed random variables with mean zero and variance one. The two cases of ρjr, the correlation coeffiecient between the j-th and r-th coefficients, being either (i) constant, or (ii) ρ∣j−r∣ρ, j ≠ r, 0 < ρ < 1, are considered. It is shown that the previous result for ρjr = 0 still remains valid for both cases.

scholarly journals On real zeros of random polynomials with hyperbolic elements

1998 ◽  
Vol 21 (2) ◽  
pp. 347-350
Author(s):  
K. Farahmand ◽  
M. Jahangiri
Keyword(s):  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Polynomials ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Normally Distributed

This paper provides the asymptotic estimate for the expected number of real zeros of a random hyperbolic polynomialg1coshx+2g2cosh2x+…+ngncoshnxwheregj,(j=1,2,…,n)are independent normally distributed random variables with mean zero and variance one. It is shown that for sufficiently largenthis asymptotic value is(1/π)logn.

scholarly journals Real zeros of a random polynomial with Legendre elements

1997 ◽  
Vol 10 (3) ◽  
pp. 257-264
Author(s):  
K. Farahmand
Keyword(s):  
Legendre Polynomials ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Polynomial ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Normally Distributed

Let T0∗(x),T1∗(x),…,Tn∗(x) be a sequence of normalized Legendre polynomials orthogonal with respect to the interval (−1,1). The asymptotic estimate of the expected number of real zeros of the random polynomial g0T0∗(x)+g1T1∗(x)+…+gnTn∗(x) where gj, j=1,2,…,n are independent identically and normally distributed random variables with mean zero and variance one is known. The present paper considers the case when the means and variances of the coefficients are not all necessarily equal. It is shown that in general this expected number of real zeros is only dependent on variances and is independent of the means.

scholarly journals Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mina Ketan Mahanti ◽  
Amandeep Singh ◽  
Lokanath Sahoo
Keyword(s):  
Random Variables ◽  
Random Coefficients ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Gaussian Random Variables ◽  
Hyperbolic Polynomials ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the formy=Pnt=n1a1cosh⁡t+n2a2cosh⁡2t+⋯+nnancosh⁡nt, wherea1,…,anis a sequence of standard Gaussian random variables, isn/2+op(1). It is shown that the asymptotic value of expected number of times the polynomial crosses the levely=Kis alson/2as long asKdoes not exceed2neμ(n), whereμ(n)=o(n). The number of oscillations ofPn(t)abouty=Kwill be less thann/2asymptotically only ifK=2neμ(n), whereμ(n)=O(n)orn-1μ(n)→∞. In the former case the number of oscillations continues to be a fraction ofnand decreases with the increase in value ofμ(n). In the latter case, the number of oscillations reduces toop(n)and almost no trace of the curve is expected to be present above the levely=Kifμ(n)/(nlogn)→∞.

scholarly journals Large level crossings of a random polynomial

1988 ◽  
Vol 1 (4) ◽  
pp. 259-269 ◽  
Author(s):  
Kambiz Farahmand
Keyword(s):  
Random Polynomial ◽  
Random Real ◽  
Expected Number ◽  
Level Crossings ◽  
Large Level ◽  
Real Value ◽  
Normally Distributed

We know the expected number of times that a polynomial of degree n with independent random real coefficients asymptotically crosses the level K, when K is any real value such that (K2/n)→0 as n→∞. The present paper shows that, when K is allowed to be large, this expected number of crossings reduces to only one. The coefficients of the polynomial are assumed to be normally distributed. It is shown that it is sufficient to let K≥exp(nf) where f is any function of n such that f→∞ as n→∞.

scholarly journals Algebraic polynomials with random coefficients

2002 ◽  
Vol 15 (1) ◽  
pp. 83-88
Author(s):  
K. Farahmand
Keyword(s):  
Probability Space ◽  
Random Variables ◽  
Random Coefficients ◽  
Random Polynomial ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Asymptotic Value ◽  
Fixed Probability ◽  
Special Case ◽  
Normally Distributed

This paper provides an asymptotic value for the mathematical expected number of points of inflections of a random polynomial of the form a0(ω)+a1(ω)(n1)1/2x+a2(ω)(n2)1/2x2+…an(ω)(nn)1/2xn when n is large. The coefficients {aj(w)}j=0n, w∈Ω are assumed to be a sequence of independent normally distributed random variables with means zero and variance one, each defined on a fixed probability space (A,Ω,Pr). A special case of dependent coefficients is also studied.

scholarly journals Zeros of an algebraic polynomial with nonequal means random coefficients

2004 ◽  
Vol 2004 (63) ◽  
pp. 3389-3395
Author(s):  
K. Farahmand ◽  
P. Flood
Keyword(s):  
Algebraic Polynomial ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Coefficients ◽  
Random Polynomial ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
The Mean

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomiala0+a1x+a2x2+⋯+an−1xn−1. The coefficientsaj(j=0,1,2,…,n−1)are assumed to be independent normal random variables with nonidentical means. Previous results are mainly for identically distributed coefficients. Our result remains valid when the means of the coefficients are divided into many groups of equal sizes. We show that the behaviour of the random polynomial is dictated by the mean of the first group of the coefficients in the interval(−1,1)and the mean of the last group in(−∞,−1)∪(1,∞).

scholarly journals On random orthogonal polynomials

2001 ◽  
Vol 14 (3) ◽  
pp. 265-274 ◽  
Author(s):  
K. Farahmand
Keyword(s):  
Orthogonal Polynomials ◽  
Legendre Polynomials ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Polynomial ◽  
Expected Number ◽  
Dependent Random Variables ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

Let T0∗(x),T1∗(x),…,Tn∗(x) be a sequence of normalized Legendre polynomials orthogonal with respect to the interval (−1,1). The asymptotic estimate of the expected number of real zeros of the random polynomial g0T0∗(x)+g1T1∗(x)+…+gnTn∗(x) where gj, j=1,2,…,n are independent identically and normally distributed random variables is known. In this paper, we first present the asymptotic value for the above expected number when coefficients are dependent random variables. Further, for the case of independent coefficients, we define the expected number of zero up-crossings with slope greater than u or zero down-crossings with slope less than −u. Promoted by the graphical interpretation, we define these crossings as u-sharp. For the above polynomial, we provide the expected number of such crossings.

scholarly journals Mean number of real zeros of a random hyperbolic polynomial

2000 ◽  
Vol 23 (5) ◽  
pp. 335-342 ◽  
Author(s):  
J. Ernest Wilkins
Keyword(s):  
Random Variables ◽  
Hyperbolic Polynomial ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
The Mean ◽  
Normally Distributed

Consider the random hyperbolic polynomial,f(x)=1pa1coshx+⋯+np×ancoshnx, in whichnandpare integers such thatn≥2,   p≥0, and the coefficientsak(k=1,2,…,n)are independent, standard normally distributed random variables. Ifνnpis the mean number of real zeros off(x), then we prove thatνnp=π−1 logn+O{(logn)1/2}.

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