Normal approximation of stabilizing Poisson pair functionals with column-type dependence

Abstract

In this paper, we study two specific types of $d$-dimensional Poisson functionals: a double-sum type and a sum-log-sum type, both over pairs of Poisson points. On these functionals, we impose column-type dependence, i.e., local behavior in the first $k$ directions and allow non-local, yet stabilizing behavior in the remaining $d-k$ directions. The main contribution of the paper is to establish sufficient conditions for Normal approximation for sequences of such functionals over growing regions. Specifically, for any fixed region, we provide an upper bound on the Wasserstein distance between each functional and the standard Normal distribution. We then apply these results to several examples. Inspired by problems in computer science, we prove a Normal approximation for the rectilinear crossing number, arising from projections of certain random graphs onto a 2-dimensional plane. From the field of topological data analysis, we examine two types of barcode summaries, the inversion count and the tree realization number, and establish Normal approximations for both summaries under suitable models of the topological lifetimes.

Figures (10)

• Figure 1: Illustrations of the three examples highlighted in Section \ref{['sec:motivational_examples']}.
• Figure 2: The black square is the rectangle $W_n$, the light blue vertical strip is the slab $S_n^k(x,s)$, and the red dashed square is the non-stable cube $Q(x,R_n(x))$. Since $Z$ is the only Poisson point in this cube, the insertion of $x$ can only affect scores between $Z$ and another Poisson point, and not between any two Poisson points both outside this cube.
• Figure 3: Left: The white region is $W_n \setminus S(x,n^\varepsilon)$, the light blue $S(x,n^\varepsilon) \setminus Q(x,n^\varepsilon)$, and the darker blue $Q(x,n^\varepsilon)$, i.e., Cases I-III as defined in \ref{['eq:three_cases']}. Right: Illustration of the event that $R_n(x)+R_n(y) \leqslant n^\varepsilon$. When $y$ lies in either Case I or II, this implies that the two non-stable cubes around $x$ and $y$ (red dashed squares verbatim to Figure \ref{['fig:partition']}) are disjoint.
• Figure 4: Illustration of the ordering of the cubes $Q_{n,j,r}$ covering $W_n$ in the case $d=2$. All the cubes have the same volume, which doesn't depend on $n$.
• Figure 5: Illustration of projections of a random 3-dimensional geometric graph onto a green 2-dimensional plane. The crossing number counts the number of intersections between edges in the green projection plane, which is 2 for this particular graph.

Theorems & Definitions (59)

• Theorem 2.1: Normal approximation of $\Sigma$
• Theorem 2.2: Normal approximation for $\Sigma^{\log}_n$
• Corollary 2.3: Asymptotic normality of $\Pi$
• Lemma 2.4: Error bounds: Double-Sum
• Lemma 2.5: Normalization
• proof : Proof of Theorem \ref{['thm:sum_clt']}
• Lemma 2.6: Error bounds: Sum-log-sum
• proof : Proof of Theorem \ref{['thm:log_clt']}
• Lemma 3.1: Sub-polynomial moments criteria
• Lemma 3.2: Variance lower bound