Ju l 2 00 9 q - EXCHANGEABILITY VIA QUASI - INVARIANCE

For positive q, the q-exchangeability is introduced as quasi-invariance under permutations, with a special cocycle. This allows us to extend the q-analogue of de Finetti’s theorem for binary sequences [GO2] to the general real-valued sequences. In contrast to the classical case with q = 1, the order on R plays for the q-analogues a significant role. An explicit construction of ergodic q-exchangeable measures involves a random shuffling of N = {1, 2, . . .} by iteration of the geometric choice. For q distinct from 1, the shuffling yields a probability measure Q that is supported by the group of bijections of N, and has the property of quasi-invariance under both left and right multiplications by finite permutations. We establish connections of the q-exchangeability to certain transient Markov chains on the q-Pascal pyramids and to invariant random flags over the Galois fields.