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

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REZAKHAH S.; SHEMEHSAVAR S.

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ISSN: 2588-4824

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Journal: BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY Year:2011 | Volume:37 | Issue:1 Page(s): 81-92

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Title

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

Author(s)

REZAKHAH S. | SHEMEHSAVAR S. | Issue Writer Certificate

Keywords

RANDOM ALGEBRAIC POLYNOMIAL

NUMBER OF REAL ZEROS

SHARP CROSSINGS

EXPECTED DENSITY

BROWNIAN MOTION

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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APA: Copy

REZAKHAH, S., & SHEMEHSAVAR, S.. (2011). ON THE AVERAGE NUMBER OF SHARP CROSSINGS OF CERTAIN GAUSSIAN RANDOM POLYNOMIALS. BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY, 37(1), 81-92. SID. https://sid.ir/paper/579664/en

Vancouver: Copy

REZAKHAH S., SHEMEHSAVAR S.. ON THE AVERAGE NUMBER OF SHARP CROSSINGS OF CERTAIN GAUSSIAN RANDOM POLYNOMIALS. BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY[Internet]. 2011;37(1):81-92. Available from: https://sid.ir/paper/579664/en

IEEE: Copy

S. REZAKHAH, and S. SHEMEHSAVAR, “ON THE AVERAGE NUMBER OF SHARP CROSSINGS OF CERTAIN GAUSSIAN RANDOM POLYNOMIALS,” BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY, vol. 37, no. 1, pp. 81–92, 2011, [Online]. Available: https://sid.ir/paper/579664/en

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