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Information Journal Paper

Title

ON THE AVERAGE NUMBER OF SHARP CROSSINGS OF CERTAIN GAUSSIAN RANDOM POLYNOMIALS

Pages

 Start Page 81 | End Page 92

Abstract

 Let Qn(x)= Si=0n Aixi be a RANDOM ALGEBRAIC POLYNOMIAL where the coefficientsA0, A1,… form a sequence of centered Gaussian random variables. Moreover, assume that the increments Dj=Aj -Aj-1, j=0, 1, 2, …, are independent, assuming A-1=0. The coefficients can be considered as n consecutive observations of a BROWNIAN MOTION. We obtain the asymptotic behaviour of the expected number of u-sharp crossings, u>0, of polynomial Qn (x). We refer to u-sharp crossings as those zero upcrossings with slope greater than u, or those down-crossings with slope smaller than -u. We consider the cases where u is unbounded and increasing with n, say u=o (n5/4), and u=o (n3/2).

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