scholarly journals The central limit theorem for Euclidean minimal spanning trees II

1999 ◽  
Vol 31 (4) ◽  
pp. 969-984 ◽  
Author(s):  
Sungchul Lee
Keyword(s):  
Central Limit ◽  
Spanning Tree ◽  
Limit Theorems ◽  
Spanning Trees ◽  
Regularity Conditions ◽  
Minimal Spanning Tree ◽  
The Central Limit Theorem ◽  
Common Distribution ◽  
The Common ◽  
Minimal Spanning Trees

Let Xi : i ≥ 1 be i.i.d. points in ℝd, d ≥ 2, and let Tn be a minimal spanning tree on X1,…,Xn. Let L(X1,…,Xn) be the length of Tn and for each strictly positive integer α let N(X1,…,Xn;α) be the number of vertices of degree α in Tn. If the common distribution satisfies certain regularity conditions, then we prove central limit theorems for L(X1,…,Xn) and N(X1,…,Xn;α). We also study the rate of convergence for EL(X1,…,Xn).

scholarly journals The central limit theorem for Euclidean minimal spanning trees II

1999 ◽  
Vol 31 (04) ◽  
pp. 969-984 ◽  
Author(s):  
Sungchul Lee
Keyword(s):  
Central Limit ◽  
Spanning Tree ◽  
Limit Theorems ◽  
Spanning Trees ◽  
Regularity Conditions ◽  
Minimal Spanning Tree ◽  
The Central Limit Theorem ◽  
Common Distribution ◽  
The Common ◽  
Minimal Spanning Trees

Let X i : i ≥ 1 be i.i.d. points in ℝ d , d ≥ 2, and let T n be a minimal spanning tree on X 1,…,X n . Let L(X 1,…,X n ) be the length of T n and for each strictly positive integer α let N(X 1,…,X n ;α) be the number of vertices of degree α in T n . If the common distribution satisfies certain regularity conditions, then we prove central limit theorems for L(X 1,…,X n ) and N(X 1,…,X n ;α). We also study the rate of convergence for EL(X 1,…,X n ).

On the number of leaves of a euclidean minimal spanning tree

1987 ◽  
Vol 24 (4) ◽  
pp. 809-826 ◽  
Author(s):  
J. Michael Steele ◽  
Lawrence A. Shepp ◽  
William F. Eddy
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Random Variables ◽  
Minimal Spanning Tree ◽  
Absolutely Continuous ◽  
Mean And Variance ◽  
Number Of Leaves ◽  
The Mean ◽  
Vertex Degrees ◽  
Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi, , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n–1Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

On the number of leaves of a euclidean minimal spanning tree

1987 ◽  
Vol 24 (04) ◽  
pp. 809-826 ◽  
Author(s):  
J. Michael Steele ◽  
Lawrence A. Shepp ◽  
William F. Eddy
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Random Variables ◽  
Minimal Spanning Tree ◽  
Absolutely Continuous ◽  
Mean And Variance ◽  
Number Of Leaves ◽  
The Mean ◽  
Vertex Degrees ◽  
Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi , , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n –1 Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

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.

scholarly journals On a random directed spanning tree

2004 ◽  
Vol 36 (1) ◽  
pp. 19-42 ◽  
Author(s):  
Abhay G. Bhatt ◽  
Rahul Roy
Keyword(s):  
Spanning Tree ◽  
Poisson Point Process ◽  
Spanning Trees ◽  
Asymptotic Properties ◽  
Record Values ◽  
Extreme Value Statistics ◽  
Minimal Spanning Tree ◽  
Unit Square ◽  
Directed Spanning Tree ◽  
Number Of Branches

We study the asymptotic properties of a minimal spanning tree formed by n points uniformly distributed in the unit square, where the minimality is amongst all rooted spanning trees with a direction of growth. We show that the number of branches from the root of this tree, the total length of these branches, and the length of the longest branch each converges weakly. This model is related to the study of record values in the theory of extreme-value statistics and this relation is used to obtain our results. The results also hold when the tree is formed from a Poisson point process of intensity n in the unit square.

scholarly journals On a random directed spanning tree

2004 ◽  
Vol 36 (01) ◽  
pp. 19-42 ◽  
Author(s):  
Abhay G. Bhatt ◽  
Rahul Roy
Keyword(s):  
Spanning Tree ◽  
Poisson Point Process ◽  
Spanning Trees ◽  
Asymptotic Properties ◽  
Record Values ◽  
Extreme Value Statistics ◽  
Minimal Spanning Tree ◽  
Unit Square ◽  
Directed Spanning Tree ◽  
Number Of Branches

We study the asymptotic properties of a minimal spanning tree formed by n points uniformly distributed in the unit square, where the minimality is amongst all rooted spanning trees with a direction of growth. We show that the number of branches from the root of this tree, the total length of these branches, and the length of the longest branch each converges weakly. This model is related to the study of record values in the theory of extreme-value statistics and this relation is used to obtain our results. The results also hold when the tree is formed from a Poisson point process of intensity n in the unit square.

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