Stochastic realisers of non-degenerate full-degree Type II reduced Ito polynomials

Abstract

Let $f_{\alpha }(x)=(x^q-(1-\alpha ){)}^d-{\alpha }^d{x}^z$, where $0<\alpha <1$, $q\ge 2$, $d\ge 2$, $1\le z\le q-1$, and $\gcd (q,z)=1$. These are the non-degenerate full-degree Type II reduced Ito polynomials of order $n=qd$. We give a complete parametrisation, up to permutation similarity, of the stochastic matrices whose characteristic polynomial is $f_{\alpha }$. The support data are a composition $a_0+\cdots +a_{d-1}=z$ and a cyclic origin $\rho \in {\mathbb{Z}}_q$. The transfer support in block 0 is an arbitrary nonempty subset of $[\rho ,\rho +a_0{]}_q$, while the support in block $t\ge 1$ is an arbitrary nonempty subset of $[\rho -{\sum }_{j=1}^t(a_j+1),\rho -{\sum }_{j=1}^t(a_j+1)+a_t{]}_q$. The horizontal weights at the selected positions are arbitrary elements of $(0,1)$ whose product in every block is $1-\alpha$. The proof first shows that the order-$n$ Type II arc lies strictly outside ${\Theta }_{n-1}$, which permits the Dmitriev--Dynkin/Kirkland--Smigoc two-shift reduction. A circular separation theorem then classifies the possible supports, and Coates' formula supplies the characteristic polynomial. This resolves the non-sparse full-degree Type II characteristic-polynomial realiser problem.

arXiv comment22 pages, 2 figures