Fixed points and cycle structure of random permutations. (English) Zbl 1343.05011

Mukherjee, Sumit

Electron. J. Probab. 21, Paper No. 40, 18 p. (2016).

Summary: Using the recently developed notion of permutation limits this paper derives the limiting distribution of the number of fixed points and cycle structure for any convergent sequence of random permutations, under mild regularity conditions. In particular this covers random permutations generated from Mallows model with Kendall’s tau, \(\mu \) random permutations introduced in [C. Hoppen et al., J. Comb. Theory, Ser. B 103, No. 1, 93–113 (2013; Zbl 1255.05174)], as well as a class of exponential families introduced in [S. Mukherjee, Ann. Stat. 44, No. 2, 853–875 (2016; Zbl 1341.62083)].

MSC:

05A05	Permutations, words, matrices
60C05	Combinatorial probability
60F05	Central limit and other weak theorems
62F12	Asymptotic properties of parametric estimators
62F10	Point estimation

Keywords:

combinatorial probability; Mallows model; permutation limit; fixed points; cycle structure