Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (4) ◽  
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}, v0 = min {n : U(∊)n = M∊}, v1 = max {n : U(∊)n = M∊}. The joint limiting distribution of ∊2σ∊–2v0 and ∊σ∊–2M∊ is determined. It is the same as for ∊2σ∊–2v1 and ∊σ–2∊M∊. The marginal ∊σ–2∊M∊ gives Kingman's heavy traffic theorem. Also lim ∊–1P(M∊ = 0) and lim ∊–1P(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.

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 ∊ &lt; 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.

A Quenched Central Limit Theorem for Reversible Random Walks in a Random Environment on Z

2014 ◽  
Vol 51 (04) ◽  
pp. 1051-1064
Author(s):  
Hoang-Chuong Lam
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Integrability Condition ◽  
The Central Limit Theorem ◽  
The Poisson Equation ◽  
Convergence Of Moments ◽  
Quenched Central Limit Theorem

The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on Z without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple and was introduced by Depauw and Derrien [3]. More precisely, for a given realization ω of the environment, we consider the Poisson equation (P ω - I)g = f, and then use the pointwise ergodic theorem in [8] to treat the limit of solutions and then the central limit theorem will be established by the convergence of moments. In particular, there is an analogue to a Markov process with discrete space and the diffusion in a stationary random environment.

scholarly journals A note on the central limit theorem for geodesic random walks

1984 ◽  
Vol 30 (2) ◽  
pp. 169-173 ◽  
Author(s):  
Gilles Blum
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
The Central Limit Theorem

In this article we use a theorem of T.G. Kurtz to prove a central limit theorem for geodesic random walks.

The central limit theorem for m-dependent variables

1955 ◽  
Vol 51 (1) ◽  
pp. 92-95 ◽  
Author(s):  
P. H. Diananda
Keyword(s):  
Standard Deviation ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Sufficient Conditions ◽  
The Central Limit Theorem ◽  
Lindeberg Condition ◽  
Dependent Variables ◽  
Basic Ideas

In a previous paper (4) central limit theorems were obtained for sequences of m-dependent random variables (r.v.'s) asymptotically stationary to second order, the sufficient conditions being akin to the Lindeberg condition (3). In this paper similar theorems are obtained for sequences of m-dependent r.v.'s with bounded variances and with the property that for large n, where s′n is the standard deviation of the nth partial sum of the sequence. The same basic ideas as in (4) are used, but the proofs have been simplified. The results of this paper are examined in relation to earlier ones of Hoeffding and Robbins(5) and of the author (4). The cases of identically distributed r.v.'s and of vector r.v.'s are mentioned.

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