Are random permutations spherically uniform?
نویسندگان
چکیده
منابع مشابه
Spherically Symmetric Random Permutations
We consider random permutations which are spherically symmetric with respect to a metric on the symmetric group Sn and are consistent as n varies. The extreme infinitely spherically symmetric permutation-valued processes are identified for the Hamming, Kendall-tau and Caley metrics. The proofs in all three cases are based on a unified approach through stochastic monotonicity. MSC:
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Proof Write Ck as the sum of indicator random variables 1γ , for γ a k-cycle. This means that 1γ(π) is 1 if γ is a cycle of π and 0 otherwise. Then E(Ck) = ∑ γ E(1γ). To determine E(1γ) we count the number of permutations having γ as a cycle. That number is (n− k)!. Thus, E(1γ) = (n − k)!/n!. Now, the number of possible γ is n(n − 1) · · · (n − k + 1)/k, since a k-cycle is an ordered selection ...
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ژورنال
عنوان ژورنال: Electronic Journal of Probability
سال: 2020
ISSN: 1083-6489
DOI: 10.1214/20-ejp418