A Generalization of the Erdős–Turán Law for the Order of Random Permutation

2012 ◽  
Vol 21 (5) ◽  
pp. 715-733 ◽  
Author(s):  
ALEXANDER GNEDIN ◽  
ALEXANDER IKSANOV ◽  
ALEXANDER MARYNYCH
Keyword(s):  
Markov Chain ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Random Permutation ◽  
Unit Interval ◽  
Random Permutations ◽  
Cycle Structure ◽  
Limit Laws ◽  
Cycle Lengths

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on n integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erdős–Turán law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM(θ)-distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths.

scholarly journals A Self-Normalized Central Limit Theorem for Markov Random Walks

2012 ◽  
Vol 44 (02) ◽  
pp. 452-478
Author(s):  
Cheng-Der Fuh ◽  
Tian-Xiao Pang
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
State Space ◽  
Central Limit ◽  
Infinite Variance ◽  
Markov Random ◽  
Markov Random Walk

Motivated by the study of the asymptotic normality of the least-squares estimator in the (autoregressive) AR(1) model under possibly infinite variance, in this paper we investigate a self-normalized central limit theorem for Markov random walks. That is, let {X n , n ≥ 0} be a Markov chain on a general state space X with transition probability P and invariant measure π. Suppose that an additive component S n takes values on the real line , and is adjoined to the chain such that {S n , n ≥ 1} is a Markov random walk. Assume that S n = ∑ k=1 n ξ k , and that {ξ n , n ≥ 1} is a nondegenerate and stationary sequence under π that belongs to the domain of attraction of the normal law with zero mean and possibly infinite variance. By making use of an asymptotic variance formula of S n / √n, we prove a self-normalized central limit theorem for S n under some regularity conditions. An essential idea in our proof is to bound the covariance of the Markov random walk via a sequence of weight functions, which plays a crucial role in determining the moment condition and dependence structure of the Markov random walk. As illustrations, we apply our results to the finite-state Markov chain, the AR(1) model, and the linear state space model.

scholarly journals A Self-Normalized Central Limit Theorem for Markov Random Walks

2012 ◽  
Vol 44 (2) ◽  
pp. 452-478 ◽  
Author(s):  
Cheng-Der Fuh ◽  
Tian-Xiao Pang
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
State Space ◽  
Central Limit ◽  
Infinite Variance ◽  
Markov Random ◽  
Markov Random Walk

Motivated by the study of the asymptotic normality of the least-squares estimator in the (autoregressive) AR(1) model under possibly infinite variance, in this paper we investigate a self-normalized central limit theorem for Markov random walks. That is, let {Xn, n ≥ 0} be a Markov chain on a general state space X with transition probability P and invariant measure π. Suppose that an additive component Sn takes values on the real line , and is adjoined to the chain such that {Sn, n ≥ 1} is a Markov random walk. Assume that Sn = ∑k=1nξk, and that {ξn, n ≥ 1} is a nondegenerate and stationary sequence under π that belongs to the domain of attraction of the normal law with zero mean and possibly infinite variance. By making use of an asymptotic variance formula of Sn / √n, we prove a self-normalized central limit theorem for Sn under some regularity conditions. An essential idea in our proof is to bound the covariance of the Markov random walk via a sequence of weight functions, which plays a crucial role in determining the moment condition and dependence structure of the Markov random walk. As illustrations, we apply our results to the finite-state Markov chain, the AR(1) model, and the linear state space model.

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment

2004 ◽  
Vol 41 (01) ◽  
pp. 83-92 ◽  
Author(s):  
Jean Bérard
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Space Time ◽  
Nearest Neighbour ◽  
One Dimensional ◽  
The Central Limit Theorem ◽  
Almost All

The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabeiet al.for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for anarbitrarylevel of randomness.

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