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

TL;DR

This paper investigates the average number of real roots of random polynomials with coefficients derived from fractional Brownian motion increments, revealing dependence on the Hurst index and extending classical results.

Contribution

It generalizes Kac's classical result to correlated coefficients from fractional Brownian motion, showing how the Hurst index influences the expected number of real zeros.

Findings

Average number of real zeros grows as ~K_H log n for large n

Hurst index H affects only the count of positive zeros

For H and 1-H, the number of real zeros are essentially the same

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.