Expected Number of Slope Crossings of Certain Gaussian Random Polynomials

Abstract

Let be a random polynomial where the coefficients A ₀, A ₁,… form a sequence of centered Gaussian random variables. Moreover, assume that the increments Δ _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 study the number of times that such a random polynomial crosses a line which is not necessarily parallel to the x-axis. More precisely we obtain the asymptotic behavior of the expected number of real roots of the equation Q _n (x) = Kx, for the cases that K is any non-zero real constant K = o(n ^1/4), and K = o(n ^1/2) separately.

Keywords:

• Brownian motion
• Expected density
• Number of real zeros
• Random algebraic polynomial
• Slope crossing

Mathematics Subject Classification:

• Primary 60H42
• Secondary 60G99

The authors would like to thank the referee for reading carefully the manuscript and providing valuable comments.