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.

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.

The average number of real zeros of a random trigonometric polynomial

1968 ◽  
Vol 64 (3) ◽  
pp. 721-730 ◽  
Author(s):  
Minaketan Das
Keyword(s):  
Trigonometric Polynomial ◽  
Mathematical Expectation ◽  
Random Variables ◽  
Trigonometric Polynomials ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
Image Position ◽  
Mutually Independent

AbstractLet a1, a2,… be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity; let b1, b2,… be a set of positive constants. In this work, we obtain the average number of zeros in the interval (0, 2π) of trigonometric polynomials of the formfor large n. The case when bk = kσ (σ > − 3/2;) is studied in detail. Here the required average is (2σ + 1/2σ + 3)½.2n + o(n) for σ ≥ − ½ and of order n3/2; + σ in the remaining cases.

scholarly journals Mean number of real zeros of a random trigonometric polynomial IV

1997 ◽  
Vol 10 (1) ◽  
pp. 67-70 ◽  
Author(s):  
J. Ernest Wilkins
Keyword(s):  
Positive Integer ◽  
Trigonometric Polynomial ◽  
Random Variables ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
The Mean ◽  
Normally Distributed

If aj(j=1,2,…,n) are independent, normally distributed random variables with mean 0 and variance 1, if p is one half of any odd positive integer except one, and if vnp is the mean number of zeros on (0,2π) of the trigonometric polynomial a1cosx+2pa2cos2x+…+npancosnx, then vnp=μp{(2n+1)+D1p+(2n+1)−1D2p+(2n+1)−2D3p}+O{(2n+1)−3}, in which μp={(2p+1)/(2p+3)}½, and D1p, D2p and D3p are explicitly stated constants.

scholarly journals Covariance of the number of real zeros of a random trigonometric polynomial

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
K. Farahmand ◽  
M. Sambandham
Keyword(s):  
Trigonometric Polynomial ◽  
Random Coefficients ◽  
Expected Number ◽  
Asymptotic Value ◽  
Advanced Method ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial

For random coefficientsajandbjwe consider a random trigonometric polynomial defined asTn(θ)=∑j=0n{ajcos⁡jθ+bjsin⁡jθ}. The expected number of real zeros ofTn(θ)in the interval(0,2π)can be easily obtained. In this note we show that this number is in factn/3. However the variance of the above number is not known. This note presents a method which leads to the asymptotic value for the covariance of the number of real zeros of the above polynomial in intervals(0,π)and(π,2π). It can be seen that our method in fact remains valid to obtain the result for any two disjoint intervals. The applicability of our method to the classical random trigonometric polynomial, defined asPn(θ)=∑j=0naj(ω)cos⁡jθ, is also discussed.Tn(θ)has the advantage onPn(θ)of being stationary, with respect toθ, for which, therefore, a more advanced method developed could be used to yield the results.

scholarly journals Mean number of real zeros of a random trigonometric polynomial. III

1995 ◽  
Vol 8 (3) ◽  
pp. 299-317
Author(s):  
J. Ernest Wilkins ◽  
Shantay A. Souter
Keyword(s):  
Trigonometric Polynomial ◽  
Approximate Formula ◽  
Error Term ◽  
Random Variables ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
The Mean ◽  
Normally Distributed

If a1,a2,…,an are independent, normally distributed random variables with mean 0 and variance 1, and if vn is the mean number of zeros on the interval (0,2π) of the trigonometric polynomial a1cosx+2½a2cos2x+…+n½ancosnx, then vn=2−½{(2n+1)+D1+(2n+1)−1D2+(2n+1)−2D3}+O{(2n+1)−3}, in which D1=−0.378124, D2=−12, D3=0.5523. After tabulation of 5D values of vn when n=1(1)40, we find that the approximate formula for vn, obtained from the above result when the error term is neglected, produces 5D values that are in error by at most 10−5 when n≥8, and by only about 0.1% when n=2.

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 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 Real zeros of random algebraic polynomials with binomial elements

2006 ◽  
Vol 2006 ◽  
pp. 1-6 ◽  
Author(s):  
A. Nezakati ◽  
K. Farahmand
Keyword(s):  
Algebraic Polynomial ◽  
Asymptotic Estimate ◽  
General Theorem ◽  
Random Variables ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Random Algebraic Polynomials ◽  
Special Case

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomial a0+a1x+a2x2+…+an−1xn−1. The coefficients aj(j=0,1,2,…,n−1) are assumed to be independent normal random variables with mean zero. For integers m and k=O(log⁡n)2 the variances of the coefficients are assumed to have nonidentical value var⁡(aj)=(k−1j−ik), where n=k⋅m and i=0,1,2,…,m−1. Previous results are mainly for identically distributed coefficients or when var⁡(aj)=(nj). We show that the latter is a special case of our general theorem.

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 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.

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