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November 2010 q-exchangeability via quasi-invariance
Alexander Gnedin, Grigori Olshanski
Ann. Probab. 38(6): 2103-2135 (November 2010). DOI: 10.1214/10-AOP536

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.

Citation

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Alexander Gnedin. Grigori Olshanski. "q-exchangeability via quasi-invariance." Ann. Probab. 38 (6) 2103 - 2135, November 2010. https://doi.org/10.1214/10-AOP536

Information

Published: November 2010
First available in Project Euclid: 24 September 2010

zbMATH: 1204.60029
MathSciNet: MR2683626
Digital Object Identifier: 10.1214/10-AOP536

Subjects:
Primary: 60C05 , 60G09
Secondary: 37A50

Keywords: Ergodic decomposition , Mallows distribution , q-exchangeability

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 6 • November 2010
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