Some bounds for the expected number of level crossings of symmetric harmonizable p-stable processes

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.

Download Full-text

Level crossings of a random trigonometric polynomial with dependent coefficients

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700038076 ◽

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.

Download Full-text

Level crossings and turning points of random hyperbolic polynomials

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171299225793 ◽

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.

Download Full-text

Large level crossings of a random polynomial

Journal of Applied Mathematics and Simulation ◽

10.1155/s104895338800019x ◽

1988 ◽

Vol 1 (4) ◽

pp. 259-269 ◽

Cited By ~ 1

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→∞.

Download Full-text