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Poisson approximation for stochastic processes summed over amenable groups

Haoyu Ye, Peter Orbanz, Morgane Austern

Abstract

We generalize the Poisson limit theorem to binary functions of random objects whose law is invariant under the action of an amenable group. Examples include stationary random fields, exchangeable sequences, and exchangeable graphs. A celebrated result of E. Lindenstrauss shows that normalized sums over certain increasing subsets of such groups approximate expectations. Our results clarify that the corresponding unnormalized sums of binary statistics are asymptotically Poisson, provided suitable mixing conditions hold. They extend further to randomly subsampled sums and also show that strict invariance of the distribution is not needed if the requisite mixing condition defined by the group holds. We illustrate the results with applications to random fields, Cayley graphs, and Poisson processes on groups.

Poisson approximation for stochastic processes summed over amenable groups

Abstract

We generalize the Poisson limit theorem to binary functions of random objects whose law is invariant under the action of an amenable group. Examples include stationary random fields, exchangeable sequences, and exchangeable graphs. A celebrated result of E. Lindenstrauss shows that normalized sums over certain increasing subsets of such groups approximate expectations. Our results clarify that the corresponding unnormalized sums of binary statistics are asymptotically Poisson, provided suitable mixing conditions hold. They extend further to randomly subsampled sums and also show that strict invariance of the distribution is not needed if the requisite mixing condition defined by the group holds. We illustrate the results with applications to random fields, Cayley graphs, and Poisson processes on groups.
Paper Structure (32 sections, 41 theorems, 306 equations)

This paper contains 32 sections, 41 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. Related literature
  3. Preliminaries
  4. Groups and amenability
  5. Stein's method for compound Poisson variables
  6. Mixing
  7. $\Psi$-mixing
  8. $\xi$-mixing
  9. Results
  10. Moment conditions
  11. Main result
  12. A simple case: Ergodicity
  13. Generalizations
  14. Applications
  15. Random fields
  16. ...and 17 more sections

Key Result

Lemma 2

Require ${\sum_j\lambda(j)<\infty}$. For every bounded function ${g: \mathbb{N} \rightarrow \mathbb{R}}$, there is a bounded function $f: \mathop{\mathrm{\mathbb{N}}}\nolimits \rightarrow \mathop{\mathrm{\mathbb{R}}}\nolimits$ that solves the equation if and only if $\mathop{\mathrm{\mathbb{E}}}\nolimits[\,g(Z(\lambda))]=0$.

Theorems & Definitions (82)

  • Example 1
  • Lemma 2: *Barbour1992CPStein
  • Lemma 3: *Barbour1992CPStein
  • Lemma 4: *Barbour1992CPStein
  • Example 5
  • Theorem 6
  • Remark
  • Corollary 7
  • Theorem 8
  • Remark
  • ...and 72 more