Average number of real roots of random polynomials defined by the increments of fractional Brownian motion

Abstract

The study of random polynomials has a long and rich history. This paper studies random algebraic polynomials $P_n(x) = a_0 + a_1 x + \ldots + a_{n-1} x^{n-1}$ where the coefficients $(a_k)$ are correlated random variables taken as the increments $X(k+1) - X(k)$, $k\in \mathbb{N}$, of a fractional Brownian motion $X$ of Hurst index $0< H < 1$. This reduces to the classical setting of independent coefficients for $H = 1/2$. We obtain that the average number of the real zeros of $P_n(x)$ is~$\sim K_H \log n$, for large $n$, where $K_H = (1 + 2 \sqrt{H(1-H)})/\pi$ (a generalisation of a classical result obtained by Kac in 1943). Unexpectedly, the parameter $H$ affects only the number of positive zeros, and the number of real zeros of the polynomials corresponding to fractional Brownian motions of indexes $H$ and $1-H$ are essentially the same. The limit case $H = 0$ presents some particularities: the average number of positive zeros converges to a constant. These results shed some light on the nature of fractional Brownian motion on the one hand and on the behaviour of real zeros of random polynomials of dependent coefficients on the other hand.