Rate of convergence for the Euclidean minimum spanning tree limit law

Jaillet, Patrick

doi:10.1016/0167-6377(93)90098-2

Cited by 9 publications

(3 citation statements)

References 9 publications

(20 reference statements)

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“…Proof of Theorem 3. Theorem 3 follows from Jaillet (1993) and Kesten and Lee (1996). Jaillet (1993) showed that there exists a strictly positive but finite constant C such that for large n EL(P 1 (n))…”

Section: Kl (λ)mentioning

confidence: 99%

The central limit theorem for Euclidean minimal spanning trees II

Lee

1999

Advances in Applied Probability

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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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Section: Kl (λ)mentioning

confidence: 99%

The central limit theorem for Euclidean minimal spanning trees II

Lee

1999

Advances in Applied Probability

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“…Our results are stated below. But, first we would like to spell out the restrictions on the Euclidean functional L. We call L(A, B, p), A a finite subset of a 4) and for a partition…”

Section: Introductionmentioning

confidence: 99%

“…Note that the bounds in Theorem 1 follow from the assumptions of the Euclidean functionals. If we use the specific property of the Euclidean functional, in some cases we can get much better bounds and surprisingly we can even get a strictly positive lower bound for large n; see Jaillet (1993), Rhee (1994), Yukich (1998), Lee (2000) for these specific results.…”

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confidence: 99%

Rates of Convergence of Means of Euclidean Functionals

Koo

Lee

2007

J Theor Probab

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Let L be the Euclidean functional with p-th power-weighted edges. Examples include the sum of the p-th power-weighted lengths of the edges in minimal spanning trees, traveling salesman tours, and minimal matchings. Motivated by the works of Steele, Yukich (1994, 1996) have shown that for n i.i.d. sample points {X 1 , . . . , X n } from [0, 1] d , L({X 1 , . . . , X n })/n (d−p)/d converges a.s. to a finite constant. Here we bound the rate of convergence of EL({X 1 , . . . , X n })/n (d−p)/d .

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