Approximating Multiples of Strong Rayleigh Random Variables

Thomas Liggett University of California, Los Angeles (UCLA) Mathematics

A polynomial with positive coefficients is said to be strong Raleigh (SR) if all of its roots are real, and hence negative. A random variable $X$ with values $0,1,\dots, n$ is said to be SR if its generating polynomial is SR. Motivated by an attempt to prove a multivariate CLT for SR random vectors, Ghosh, Liggett and Pemantle (2017) raised the question of the extent to which $\frac jkX$ can be well approximated by an SR random variable when $X$ is SR. Using the technique of polynomials with interlacing roots, we proved in that paper that $\lfloor\frac 1kX\rfloor$ is such an approximation if $j=1$. It turns out that $\lfloor\frac 2kX\rfloor$ is very far from being SR. Nevertheless, I will show that it satisfies an equally useful property known as Hurwitz. I will then speculate about corresponding properties of $\lfloor\frac jkX\rfloor$ when $j\geq 3$. In particular I will consider two families of properties $P_j$ and $Q_j$ for $j\geq 1$. For these properties, $P_1=Q_1=SR$ and $P_2=Q_2=$Hurwitz. Unfortunately, $P_3\neq Q_3$, but we can prove that $\lfloor\frac jkX\rfloor$ is $Q_j$. The more useful property is $P_j$.