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 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 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 On the average number of real zeros of a class of random algebraic equations

1990 ◽  
Vol 49 (1) ◽  
pp. 149-160 ◽  
Author(s):  
N. N. Nayak ◽  
S. Bagh
Keyword(s):  
Mathematical Expectation ◽  
Random Variables ◽  
Constant Independent ◽  
Algebraic Equations ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Mutually Independent ◽  
Normally Distributed

AbstractLet g1, g2, …, gn be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity. In this work, we obtain the average number of real zeros of the random algebraic equations Σnk=1 Kσ gk(ω)tk = C, where C is a constant independent of t and not necessarily zero. This average is (1/π) log n, when n is large and σ is non-negative.

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.

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