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Functional limit laws for the intensity measure of point processes and applications

Giacomo Francisci, Anand N. Vidyashankar

Abstract

Motivated by applications to the study of depth functions for tree-indexed random variables generated by point processes, we describe functional limit theorems for the intensity measure of point processes. Specifically, we establish uniform laws of large numbers and uniform central limit theorems over a class of bounded measurable functions for estimates of the intensity measure. Using these results, we derive the uniform asymptotic properties of half-space depth and, as corollaries, obtain the asymptotic behavior of medians and other quantiles of the standardized intensity measure. Additionally, we obtain uniform concentration upper bound for the estimator of half-space depth. As a consequence of our results, we also derive uniform consistency and uniform asymptotic normality of Lotka-Nagaev and Harris-type estimators for the Laplace transform of the point processes in a branching random walk.

Functional limit laws for the intensity measure of point processes and applications

Abstract

Motivated by applications to the study of depth functions for tree-indexed random variables generated by point processes, we describe functional limit theorems for the intensity measure of point processes. Specifically, we establish uniform laws of large numbers and uniform central limit theorems over a class of bounded measurable functions for estimates of the intensity measure. Using these results, we derive the uniform asymptotic properties of half-space depth and, as corollaries, obtain the asymptotic behavior of medians and other quantiles of the standardized intensity measure. Additionally, we obtain uniform concentration upper bound for the estimator of half-space depth. As a consequence of our results, we also derive uniform consistency and uniform asymptotic normality of Lotka-Nagaev and Harris-type estimators for the Laplace transform of the point processes in a branching random walk.
Paper Structure (10 sections, 19 theorems, 163 equations)

This paper contains 10 sections, 19 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications
  4. Depth functions
  5. Tree-indexed random variables
  6. Mixed binomial point processes
  7. Preliminary results
  8. Proofs of main results
  9. Proofs of other results
  10. Concluding remarks

Key Result

Theorem 1

Assume H1-H2. Then, $\lVert\mu_{n} - \mu \rVert \xrightarrow[]{a.s.} 0$ if one of the following conditions hold: (i) for all $\epsilon>0$ and some $p \geq 1$$\frac{1}{n} \log(N(\mathcal{F}, e_{n,p}, \epsilon)) \xrightarrow[]{p^{\ast}} 0$, or (ii) for all $\delta>0$

Theorems & Definitions (35)

  • Definition 1
  • Theorem 1
  • Proposition 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Theorem 3
  • Proposition 2
  • Lemma 1
  • Proposition 3
  • ...and 25 more