TY - JOUR
A2 - Abolnikov, Lev
AU - Farahmand, K.
AU - Grigorash, A.
AU - McGuinness, B.
PY - 2008
DA - 2008/05/22
TI - On Different Classes of Algebraic Polynomials with Random Coefficients
SP - 189675
VL - 2008
AB - The expected number of real zeros of the polynomial of the form a0+a1x+a2x2+⋯+anxn, where a0,a1,a2,…,an is a sequence of standardGaussian random variables, is known. For n large it is shown that this expectednumber in (−∞,∞) is asymptotic to (2/π)log n. In this paper, we show thatthis asymptotic value increases significantly to n+1 when we consider apolynomial in the form a0(n0)1/2x/1+a1(n1)1/2x2/2+a2(n2)1/2x3/3+⋯+an(nn)1/2xn+1/n+1 instead. We give the motivation for our choice ofpolynomial and also obtain some other characteristics for the polynomial, suchas the expected number of level crossings or maxima. We note, and present,a small modification to the definition of our polynomial which improves ourresult from the above asymptotic relation to the equality.
SN - 2090-3332
UR - https://doi.org/10.1155/2008/189675
DO - 10.1155/2008/189675
JF - Journal of Applied Mathematics and Stochastic Analysis
PB - Hindawi Publishing Corporation
KW -
ER -