2009/07/31 by Alexander Gnedin, Grigori Olshanski · 41 citations
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Contrast (vision) #Ergodic theory #Invariant (physics) #Markov chain #Markov process #Random Matrices and Applications #Random compact set #Random permutation #Shuffling #math.CO #math.PR #semigroups and automata theory
paper · pdf · doi:10.1214/10-aop536
published in The Annals of Probability 38(6) (Institute of Mathematical Statistics) · Published in at http://dx.doi.org/10.1214/10-AOP536 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2010/09/24 · arxiv created 2010/11/10 · arxiv updated 2010/11/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
For positive q ≠ 1, the q-exchangeability of an infinite random word is introduced as quasi-invariance under permutations of letters, with a special cocycle which accounts for inversions in the word. This framework allows us to extend the q-analog of de Finetti’s theorem for binary sequences—see Gnedin and Olshanski [Electron. J. Combin. 16 (2009) R78]—to general real-valued sequences. In contrast to the classical case of exchangeability (q = 1), the order on ℝ plays a significant role for the q-analogs. An explicit construction of ergodic q-exchangeable measures involves random shuffling of ℕ = 1, 2, … by iteration of the geometric choice. Connections are established with transient Markov chains on q-Pascal pyramids and invariant random flags over the Galois fields.