Expected Number of Local Maxima of Some Gaussian Random Polynomials

Details Expected Number of Local Maxima of Some Gaussian Random Polynomials Expected Number of Local Maxima of Some Gaussian Random Polynomials

2006-05-04

arxiv.org/abs/math/0605116v1

Mathematics

Probability

12 pages

Scientific paper

Let $Q_n(x)=\sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraic polynomial where the coefficients $A_0,A_1,... $ form a sequence of centered Gaussian random variables. Moreover, assume that the increments $\Delta_j=A_j-A_{j-1}$, $j=0,1,2,...$ are independent, $A_{-1}=0$. The coefficients can be considered as $n$ consecutive observations of a Brownian motion. We study the asymptotic behaviour of the expected number of local maxima of $Q_n(x)$ below level $u=O(n^k)$, for some $k>0$.

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