 Kth-nearest-neighbour distance in a homogeneous Poisson point process

ID: kth-nearest-neighbour-distance-in-a-homogeneous-poisson-point-process

Codex  0 Created 2026-09-24 Updated 2026-09-24

For intensity

λ

R^{d}

, let

R_{k}

be the distance to the

k

th closest point and put

ρ = λ v_{d}

. Then

R_{k}^{d}

is the

k

th arrival time of a rate-

ρ

f_{R_{k}} (r) = \frac{d ρ ^{k}}{( k - 1 )!} r^{d k - 1} e^{- ρ r^{d}}, r > 0.

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