On asymptotic expansions for the distributions of the number of cycles in a random permutation
-
A. N. Timashov
We obtain explicit formulas for the coefficients of asymptotic expansions in the domain of large deviations for the distributions of the number of cycles νn in a random permutation of degree n, that is, for the probability P {νn = N} under the condition that n, N → ∞ in such a way that 1 < α0 ≤ α = n / N ≤ α1 < ∞, where α0, α1 are constants. These formulas express the coefficients in terms of cumulants of the random variable which has the distribution of the logarithmic series with specially chosen parameter. For the cumulants of the third and fourth orders we give the corresponding values. We discuss the accuracy of the obtained approximations. If n, N → ∞ so that 0 < γ0 ≤ γ = N / ln n ≤ γ1 < ∞, where γ0, γ1 are constants, we give asymptotic estimates of the probabilities P {νn = N}, P {νn ≤ N}, P {νn ≥ N} with the remainder terms of order O ((ln n)-2) uniform in γ ∈ [γ0,γ1]. The corresponding estimate for the probability P {νn = N} refines the previously known results for the case N = β ln n + o (ln n), where β is a positive constant.
Copyright 2003, Walter de Gruyter
Articles in the same Issue
- Boolean reducibility
- On the complexity of testing primality by homogeneous structures
- On distinguishability of states of automata
- On glueing states of an automaton
- Almost layer-finiteness of the periodic part of groups without involutions
- The structure and methods of generation of closed classes of graphs
- On asymptotic expansions for the distributions of the number of cycles in a random permutation
Articles in the same Issue
- Boolean reducibility
- On the complexity of testing primality by homogeneous structures
- On distinguishability of states of automata
- On glueing states of an automaton
- Almost layer-finiteness of the periodic part of groups without involutions
- The structure and methods of generation of closed classes of graphs
- On asymptotic expansions for the distributions of the number of cycles in a random permutation
