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The cycle structure of random permutations. (English) Zbl 0759.60007

Authors’ abstract: The total variation distance between the process which counts cycles of size \(1,2,\dots,b\) of a random permutation of \(n\) objects and a process \((Z_ 1,Z_ 2,\dots,Z_ b)\) of independent Poisson random variables with \(EZ_ i=1/i\) converges to 0 if and only if \(b/n\to 0\). This Poisson approximation can be used to give simple proofs of limit theorems and bounds for a wide variety of functionals of random permutations. These limit theorems include the Erdős-Turán theorem for the asymptotic normality of the log of the order of a random permutation, and the DeLaurentis-Pittel functional central limit theorem for the cycle sizes. We give a simple explicit upper bound on the total variation distance to show that this distance decays to zero superexponentially fast as a function of \(n/b\to\infty\). A similar result holds for derangements and, more generally, for permutations conditioned to have given number of cycles of various sizes. Comparison results are included to show that in approximating the cycle structure by an independent Poisson process the main discrepancy arises from independence rather than from Poisson marginals.

MSC:

60C05 Combinatorial probability
60F17 Functional limit theorems; invariance principles
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G18 Self-similar stochastic processes
05A05 Permutations, words, matrices
05A16 Asymptotic enumeration
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