Expected Number of Real Zeros of Lévy and Harmonizable Stable Random Polynomials

2000 ◽  
Vol 7 (2) ◽  
pp. 379-386 ◽  
Author(s):  
S. Rezakhah ◽  
A. R. Soltani
Keyword(s):  
General Formula ◽  
Algebraic Polynomial ◽  
Stable Processes ◽  
Random Polynomials ◽  
Expected Number ◽  
Explicit Formulas ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Stable Random Vector ◽  
Stable Noise

Abstract Assuming (A 0, A 1, . . . , An ) is a jointly symmetric α-stable random vector, 0 < α ≤ 2, a general formula for the expected number of real zeros of the random algebraic polynomial was obtained by Rezakhah in 1997. Rezakhah's formula is applied to the case where (A 0, . . . , An ) is formed by consecutive observations from the Lévy stable noise or from certain harmonizable stable processes, and more explicit formulas are derived for the expected number of real zeros of .

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 On the variance of the number of real zeros of a random trigonometric polynomial

1997 ◽  
Vol 10 (1) ◽  
pp. 57-66 ◽  
Author(s):  
K. Farahmand
Keyword(s):  
Trigonometric Polynomial ◽  
General Formula ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Normal Process ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
Upper Estimate

The asymptotic estimate of the expected number of real zeros of the polynomial T(θ)=g1cosθ+g2cos2θ+…+gncosnθ where gj(j=1,2,…,n) is a sequence of independent normally distributed random variables is known. The present paper provides an upper estimate for the variance of such a number. To achieve this result we first present a general formula for the covariance of the number of real zeros of any normal process, ξ(t), occurring in any two disjoint intervals. A formula for the variance of the number of real zeros of ξ(t) follows from this result.

Expected Number Of Real Zeros Of a Random Algebraic Polynomial

2018 ◽  
Vol 5 (1) ◽  
pp. 936-942
Author(s):  
Dipty Rani Dhal ◽  
DR.Prasana Kumar Mishra
Keyword(s):  
Algebraic Polynomial ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros

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 Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

2009 ◽  
Vol 2009 ◽  
pp. 1-6 ◽  
Author(s):  
K. Farahmand ◽  
M. Sambandham
Keyword(s):  
Algebraic Polynomial ◽  
Random Coefficients ◽  
Random Coefficient ◽  
Geometric Progression ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Geometric Progressions

The expected number of real zeros of an algebraic polynomial with random coefficient is known. The distribution of the coefficients is often assumed to be identical albeit allowed to have different classes of distributions. For the nonidentical case, there has been much interest where the variance of the th coefficient is . It is shown that this class of polynomials has significantly more zeros than the classical algebraic polynomials with identical coefficients. However, in the case of nonidentically distributed coefficients it is analytically necessary to assume that the means of coefficients are zero. In this work we study a case when the moments of the coefficients have both binomial and geometric progression elements. That is we assume and . We show how the above expected number of real zeros is dependent on values of and in various cases.

scholarly journals Random Trigonometric Polynomials with Nonidentically Distributed Coefficients

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
K. Farahmand ◽  
T. Li
Keyword(s):  
Closed Form ◽  
Random Variables ◽  
Expected Value ◽  
Trigonometric Polynomials ◽  
Random Polynomials ◽  
Asymptotic Estimates ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Normally Distributed

This paper provides asymptotic estimates for the expected number of real zeros of two different forms of random trigonometric polynomials, where the coefficients of polynomials are normally distributed random variables with different means and variances. For the polynomials in the form of and we give a closed form for the above expected value. With some mild assumptions on the coefficients we allow the means and variances of the coefficients to differ from each others. A case of reciprocal random polynomials for both above cases is studied.

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