On the Average Number of Sharp Crossings of Certain Gaussian Random Polynomials

Details On the Average Number of Sharp Crossings of Certain Gaussian Random Polynomials On the Average Number of Sharp Crossings of Certain Gaussian Random Polynomials

2006-05-27

arxiv.org/abs/math/0605699v1

Mathematics

Probability

11 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, 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 of polynomial $Q_n(x)$ . We refer to u-sharp crossings as those zero up-crossings with slope greater than $u$, or those down-crossings with slope smaller than $-u$. We consider the cases where $u$ is unbounded and is increasing with $n$, where $u=o(n^{5/4})$, and $u=o(n^{3/2})$ separately.

Affiliated with

Also associated with

No associations

Rating

On the Average Number of Sharp Crossings of Certain Gaussian Random Polynomials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the Average Number of Sharp Crossings of Certain Gaussian Random Polynomials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Average Number of Sharp Crossings of Certain Gaussian Random Polynomials will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-680792