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Research output: Contribution to Journal/Magazine › Journal article
Research output: Contribution to Journal/Magazine › Journal article
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TY - JOUR
T1 - The order of large random permutations with cycle weights
AU - Storm, Julia
AU - Zeindler, Dirk
PY - 2015/12/5
Y1 - 2015/12/5
N2 - The order On(σ) of a permutation σ of n objects is the smallest integer k≥1 such that the k-th iterate of σ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdös and Turán who proved in 1965 that logOn satisfies a central limit theorem. We extend this result to the so-called generalized Ewens measure in a previous paper. In this paper, we establish a local limit theorem as well as, under some extra moment condition, a precise large deviations estimate. These properties are new even for the uniform measure. Furthermore, we provide precise large deviations estimates for random permutations with polynomial cycle weights.
AB - The order On(σ) of a permutation σ of n objects is the smallest integer k≥1 such that the k-th iterate of σ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdös and Turán who proved in 1965 that logOn satisfies a central limit theorem. We extend this result to the so-called generalized Ewens measure in a previous paper. In this paper, we establish a local limit theorem as well as, under some extra moment condition, a precise large deviations estimate. These properties are new even for the uniform measure. Furthermore, we provide precise large deviations estimates for random permutations with polynomial cycle weights.
KW - random permutation
KW - order of a permutation
KW - large deviations
KW - local limit theorem
U2 - 10.1214/EJP.v20-4331
DO - 10.1214/EJP.v20-4331
M3 - Journal article
VL - 20
JO - Electronic Journal of Probability
JF - Electronic Journal of Probability
SN - 1083-6489
M1 - 126
ER -