Gaussian approximation for rooted edges in a random minimal directed spanning tree

Abstract

We study the total 𝛼-powered length of the rooted edges in a random minimal directed spanning tree -first introduced in Bhatt and Roy ( 2004) -on a Poisson process with intensity s ≥ 1 on the unit cube [0, 1] 𝑑 for 𝑑 ≥ 3. While a Dickman limit was proved in Penrose and Wade (2004) in the case of 𝑑 = 2, in dimensions three and higher, Bai, Lee and Penrose (2006) showed a Gaussian central limit theorem when 𝛼 = 1, with a rate of convergence of the order (log s) −(𝑑−2)∕4 (log log s) (𝑑+1)∕2 . In this article, we extend these results and prove a central limit theorem in any dimension 𝑑 ≥ 3 for any 𝛼 > 0. Moreover, making use of recent results in Stein's method for region-stabilizing functionals, we provide presumably optimal non-asymptotic bounds of the order (log s) −(𝑑−2)∕2 on the Wasserstein and the Kolmogorov distances between the distribution of the total 𝛼-powered length of rooted edges, suitably normalized, and that of a standard Gaussian random variable.

Gaussian approximation for sums of region-stabilizing scores

“…where for y ∈ X, M s (y) := max{M s (y) 2 , M s (y) 4 } and hs (y) := max{h s (y) 2/(4+p/2) , h s (y) 4/(4+p/2) }.…”

“…Also, (A2) holds with p = 1 and M 1,1 (x) = w(x), while (2.4) holds with r(x, y) = 4 for x ∈ Z while q(x 1 , x 2 ) ≤ 4e −4 for x 1 , x 2 ∈ Z 2 with x 2 − x 1 ∈ B + B and q(x 1 , x 2 ) = 0 otherwise. Noticing that max{w(x) 2 , w(x) 4 , w(x) 9/2 } = w(x) 2 , we obtain that for all α > 0, there exists a constant C α such that…”

“…This yields the final assertion.Proof of Theorem 2.In view of Lemma 5.5, the condition in Theorem 5.1 is satisfied with the exponent 4 + p/2 with c y := C p M s (y) 4+p/2 + h s (y)(1 + g s (y) 4 ) for y ∈ X. C 4/(4+p/2) p max M s (y) 2 , M s (y)4 + max{h s (y) 2/(4+p/2) , h s (y) 4/(4+p/2) } 1 + g s (y) C 4/(4+p/2) p G s (y),…”

“…We also allow for multiple points and for a non-uniform bound on the (4 + p)-th moment of the score functions, which is particularly important in examples involving infinite intensity measures, like stationary Poisson processes. Apart from examples presented in the current paper, further applications of our method has been elaborated in [5], where a quantitative central limit theorem is obtained for functionals of growth processes that result in generalized Johnson-Mehl tessellations, and in [4], where such a result is obtained in the context of minimal directed spanning trees in dimensions three and higher, respectively.…”

Upper large deviations for power-weighted edge lengths in spatial random networks

“…To that end, we study the upper large deviations of the sum of power-weighted edge lengths, i.e., , where the sum is taken over all network edges in a growing sampling window and denotes the power considered. This is a fundamental characteristic for spatial random networks, which has already been studied in detail for the Gilbert graph and the directed spanning forest [5, 18].…”