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q-EXCHANGEABILITY VIA QUASI-INVARIANCE

Alexander Gnedin and Grigori Olshanski
The Annals of Probability
Vol. 38, No. 6 (November 2010), pp. 2103-2135
Stable URL: http://www.jstor.org/stable/25734722
Page Count: 33
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q-EXCHANGEABILITY VIA QUASI-INVARIANCE
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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 Greschonig and Schmidt [Colloq. Math. 84/85 (2000) 495–514]—to general real-valued sequences. In contrast to the classical case of exchangeability (q = 1), the order on ${\Bbb R}$ plays a significant role for the q-analogs. An explicit construction of ergodic q-exchangeable measures involves random shuffling of ${\Bbb N}$ = {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.

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