The central limit theorem for euclidean minimal spanning trees II

Sungchul Lee

doi:10.1239/aap/1029955253

The central limit theorem for euclidean minimal spanning trees II

Sungchul Lee

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

12 Citations (Scopus)

Abstract

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 (X₁, X_n). Let L((X₁, X_n)) be the length of T_n and for each strictly positive integer α let N((X₁, . . ., 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((X₁, . . ., X_n)) and N((X₁, . . ., X_n); α). We also study the rate of convergence for EL((X₁, . . ., X_n)).

Original language	English
Pages (from-to)	969-984
Number of pages	16
Journal	Advances in Applied Probability
Volume	31
Issue number	4
DOIs	https://doi.org/10.1239/aap/1029955253
Publication status	Published - 1999

All Science Journal Classification (ASJC) codes

Statistics and Probability
Applied Mathematics

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abstract = "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)).",

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N2 - 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)).

AB - 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)).

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