q-exchangeability via quasi-invariance

By Alexander V. Gnedin, Grigori Olshanski. 2009

72 citations recorded by the source corpus.

For positive q is not 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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/898c009e158a71d9217c67ac28816fda1deca939

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Citations in the corpus

On the Cycle Structure of Mallows Permutations (2016, background): “[13, 14] and the problem of the longest increasing subsequence [21, 6, 5].”
Lengths of monotone subsequences in a Mallows permutation (2013, methodology): “The process was known to Mallows [23], and was also used by Gnedin and Olshanski [17, 18] to study variants and extensions of the Mallows measure to infinite groups of permutations.”
Representations of the Infinite Symmetric Group (2016, background): “This subject was further developed in Gnedin-Olshanski [48], [49].”