q-exchangeability via quasi-invariance

Abstract

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.