Wilson's 6-j laws and stitched Markov processes

TL;DR

This paper introduces a method to embed Wilson's 6-j laws into Markov processes, transforming them into quadratic harnesses and enabling the extension of their time domain through process stitching.

Contribution

It presents a novel approach to constructing quadratic harnesses from Wilson's 6-j laws and extends their time domain via process stitching techniques.

Findings

Constructed Markov chains with Wilson's 6-j laws using time-insertion.

Converted these chains into quadratic harnesses with classical parameter gamma.

Extended the time domain of quadratic harnesses from (0,1) to all t>0.

Abstract

We show how to insert time into the parameters of the Wilson's 6-j laws to construct discrete Markov chains with these laws. By a quadratic transformation we convert them into Markov processes with linear regressions and quadratic conditional variances. Further conversion into the "standard form" gives "quadratic harnesses" with "classical" value of parameter gamma. A random-parameter-representation of the original Markov chain allows us to stitch together two copies of the process, extending time domain of the quadratic harness from (0,1) to all t>0.