scholarly journals The expected number of level crossings of random trigonometric polynomials

2004 ◽  
Vol 17 (9) ◽  
pp. 1085-1089
Author(s):  
K. Farahmand ◽  
A. Shaposhnikov
Keyword(s):  
Trigonometric Polynomials ◽  
Expected Number ◽  
Level Crossings

scholarly journals On Zeros of Self-Reciprocal Random Algebraic Polynomials

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
K. Farahmand
Keyword(s):  
Matrix Theory ◽  
Random Coefficients ◽  
Trigonometric Polynomials ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Level Crossings ◽  
Random Algebraic Polynomials ◽  
The Relationship ◽  
Normal Standard ◽  
Independent Identically Distributed

This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial TN(θ)=∑j=0N−1{αN−jcos(j+1/2)θ+βN−jsin(j+1/2)θ}, where αj and βj, j=0,1,2,…, N−1, are sequences of independent identically distributed normal standard random variables. This type of random polynomial is produced in the study of random algebraic polynomials with complex variables and complex random coefficients, with a self-reciprocal property. We establish the relation between this type of random algebraic polynomials and the above random trigonometric polynomials, and we show that the required level crossings have the functionality form of cos(N+θ/2). We also discuss the relationship which exists and can be explored further between our random polynomials and random matrix theory.

Level crossings and stationary distributions for general dams

1980 ◽  
Vol 17 (1) ◽  
pp. 218-226 ◽  
Author(s):  
Michael Rubinovitch ◽  
J. W. Cohen
Keyword(s):  
Explicit Expression ◽  
Stationary Distribution ◽  
Basic Relation ◽  
Expected Number ◽  
Stationary Distributions ◽  
Level Crossings ◽  
Fixed Level

Level crossings in a stationary dam process with additive input and arbitrary release are considered and an explicit expression for the expected number of downcrossings (and also overcrossings) of a fixed level, per time unit, is obtained. This leads to a short derivation of a basic relation which the stationary distribution of a general dam must satisfy.

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 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 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→∞.

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