Limit laws for the diameter of a set of random points from a distribution supported by a smoothly bounded set

Abstract

We study the asymptotic behavior of the maximum interpoint distance of random points in a d-dimensional set with a unique diameter and a smooth boundary at the poles. Instead of investigating only a fixed number of n points as n tends to infinity, we consider the much more general setting in which the random points are the supports of appropriately defined Poisson processes. The main result covers the case of uniformly distributed points within a d-dimensional ellipsoid with a unique major axis. Moreover, two generalizations of the main result are established, for example a limit law for the maximum interpoint distance of random points from a Pearson type II distribution.

Appendix

Proof of Lemma 1

We only consider i = ℓ.It is clear that H_ℓissymmetric, since s^ℓis a twice continuously differentiable function. From Condition 1 we know that

$$ E \subset B_{2a}\left( (a,\mathbf{0} )\right) \qquad \text{and} \qquad E \cap \partial B_{2a}\left( (a,\mathbf{0} )\right) = \left\{ (-a,\mathbf{0} )\right\}. $$

(28)

Writing $O_{t} := \left \{ \widetilde z \in \mathbb {R}^{d-1}: |\widetilde z|<2a \right \}$ and defining the mapping $t: O_{t} \to \mathbb {R}, \widetilde z \mapsto a - \sqrt {4a^{2} - {z_{2}^{2}}- \ldots -{z_{d}^{2}}}$, the boundary of B_2a((a, 0))in {z₁ < a}can be parameterized as a hypersurface via

$$\mathbf{t}: \left\{\begin{array}{l} O_{t} \to \mathbb{R}^{d},\\ \widetilde z \mapsto \left( \ t(\widetilde z)\ ,\ \widetilde z\ \right). \end{array}\right. $$

For j, k ∈ {2,…, d}, we obtain

$$\begin{array}{@{}rcl@{}} t_{j}(\widetilde z) &=& (4a^{2} - {z_{2}^{2}} - \ldots - {z_{d}^{2}})^{-\frac{1}{2}}\cdot z_{j},\\ t_{jk}(\widetilde z) &=& (4a^{2} - {z_{2}^{2}} - \ldots - {z_{d}^{2}})^{-\frac{3}{2}}\cdot z_{j}z_{k} + (4a^{2} - {z_{2}^{2}} - \ldots - {z_{d}^{2}})^{-\frac{1}{2}}\cdot \delta_{jk}. \end{array} $$

Hence, ∇t(0) = 0, and the Hessian of t at 0 is given by $H_{t} := \frac {1}{2a}\mathrm {I}_{d-1}$. So, the second-order Taylor series expansion of t at this point has the form

$$ t(\widetilde z) = -a + \mathbf{0}^{\top}\widetilde z + \frac{1}{2}\widetilde z^{\top} H_{t} \widetilde z + R_{t}(\widetilde z), $$

(29)

where $R_{t} (\widetilde z) = o\left (|\widetilde z|^{2} \right )$. Furthermore, we have

$$ s^{\ell}(\widetilde z) = -a + \nabla s^{\ell}(\mathbf{0} )^{\top} \widetilde z + \frac{1}{2}\widetilde z^{\top} H_{\ell} \widetilde z + R_{\ell}(\widetilde z), $$

(30)

where $R_{\ell } (\widetilde z) = o\left (|\widetilde z|^{2} \right )$. In view of Eq. 28 and Condition 2, we have $t(\widetilde z) <-a + s^{\ell }(\widetilde z)$ for each $\widetilde z \in O_{\ell } \backslash \left \{ \mathbf {0} \right \}$ (observe that Eq. 28 ensures O_ℓ ⊂ O_t). Using Eqs. 29 and 30, this inequality can be rewritten as

$$-a + \frac{1}{2}\widetilde z^{\top} H_{t} \widetilde z + R_{t}(\widetilde z)< -a + \nabla s^{\ell}(\mathbf{0} )^{\top} \widetilde z + \frac{1}{2}\widetilde z^{\top} H_{\ell} \widetilde z + R_{\ell}(\widetilde z), $$

and hence

$$0< \nabla s^{\ell}(\mathbf{0} )^{\top} \widetilde z + \frac{1}{2}\widetilde z^{\top} (H_{\ell} - H_{t}) \widetilde z + \left( R_{\ell}(\widetilde z) -R_{t}(\widetilde z) \right) $$

for each $\widetilde z \in O_{\ell }\backslash \left \{ \mathbf {0} \right \}$. Since $R_{\ell } (\widetilde z)-R_{t} (\widetilde z) = o\left (|\widetilde z|^{2} \right )$, this inequality shows ∇s^ℓ(0) = 0 and that the matrix H_ℓ − H_t is positive definite. Remembering $H_{t} = \frac {1}{2a}\mathrm {I}_{d-1}$, H_ℓ has to be positive definite, too, and all eigenvalues of H_ℓ have to be larger than 1/2a. □

The following lemma shows that inequality (5) is sufficient for Condition 3:

Lemma 8

If Eq. 5 holds true, then Condition3 is fulfilled.

Proof

Inequality (5) ensures the existence of an η^∗ ∈ (0, 1) with

$$ \frac{1}{\kappa_{2}^{\ell}} + \frac{1}{{\kappa_{2}^{r}}} = 2a\eta^{*}. $$

(31)

Applying (9) (with H_ℓ and H_r replaced with the matrices D_ℓ and D_r, respectively), using (31) and some obvious transformations yield

$$\begin{array}{@{}rcl@{}} 2a\eta^{*}\left( \alpha^{\top} D_{\ell}\alpha + \beta^{\top} D_{r} \beta\right)+ 2\alpha^{\top} U_{\ell}^{\top} U_{r}\beta-|\alpha|^{2} - |\beta|^{2} \!&\ge&\! \left| \sqrt{\frac{\kappa_{2}^{\ell}}{{\kappa_{2}^{r}}}}U_{\ell}\alpha + \sqrt{\frac{{\kappa_{2}^{r}}}{\kappa_{2}^{\ell}}}U_{r}\beta \right|^{2}\\ &\ge&\! 0. \end{array} $$

Consequently, Condition 3 holds with η = η^∗, see Eq. 8. □

As mentioned before, Eq. 5 is only sufficient for the unique diameter close to the poles, not necessary. See Example 3.13 in Schrempp (2017) for an illustration of a set with unique diameter between (−a, 0) and (a, 0) for which inequality (5) is not fulfilled.