Abstract
This paper provides an asymptotic estimate for the expected number of K-level crossings of a random hyperbolic polynomial g1 coshx + g2cosh2x + ---+ gcoshnx, where gj (j = 1, 2,...,n) are independent normally distributed random variables with mean zero, variance one and K is any constant independent of x . It is shown that the result for K = 0 remains valid as long as K _ K, = O(n) .
Highlights
9t{. g/A(lwt)h}o_Ug1hbethaereseqhuaesncbeeeonf independent considerable attention given to algebraic and trigonometric polynomials with coefficients gj’s, very little is known about the behavior of the random hyperbolic polynomial, Z P(x) =_ Pn(x,w) gj()cosh jx. 3-’1
In the interval (-1,1) the hyperbolic polynomial has asymptotically as many zeros as the algebraic polynomial, outside this interval, unlike the algebraic case, the hyperbolic polynomial does not possess any sizable zeros. This could have been caused by fast increase of the terms in the hyperbolic polynomial in (- o, 1)t3 (1, oc) which makes the cancellations in this type of polynomial difficult
If one classifies the oscillation of different types of polynomials according to the behavior of their real zeros it seems interesting to note that random hyperbolic polynomials will fall into the algebraic category their properties of K-level crossings follow that of the trigonometric case
Summary
9t{. g/A(lwt)h}o_Ug1hbethaereseqhuaesncbeeeonf independent considerable attention given to algebraic and trigonometric polynomials with coefficients gj’s, very little is known about the behavior of the random hyperbolic polynomial,. LEVEL CROSSINGS OF A RANDOM POLYNOMIAL WITH HYPERBOLIC ELEMENTS Submitted for publication to the Proceedings of the American Mathematical Society Let (f,A,P) be a identically distributed fixed probability space and let random variables defined on
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