bibkey: “arratia1990poisson” authors: “Richard Arratia; Larry Goldstein; Louis Gordon” year: 1990 title: “Poisson Approximation and the Chen-Stein Method” doi: “10.1214/ss/1177012015” url: “https://doi.org/10.1214/ss/1177012015” claim: “Local-dependence terms control Poisson approximation for a count of events, including an explicit bound on its zero-count probability.” strata_touched: [] license: “citation-only” triage: “anchor”

Poisson Approximation and the Chen-Stein Method

Locator

DOI: 10.1214/ss/1177012015; canonical URL: https://doi.org/10.1214/ss/1177012015.

Statistical Science 5(4), 403–424 (1990). The dependence quantities are defined in section 3; Theorem 1 on page 406 bounds the count distribution and its zero-count probability. Theorem 2 on the same page gives a process version. The exposition attributes the proofs to the authors’ 1989 Annals of Probability paper, Two Moments Suffice for Poisson Approximations: The Chen-Stein Method.

Scope

This is literature-attested background for locally dependent rare-event counts. The parity-kernel proof supplies its own event probabilities, separation rule, factorial-moment domination, and reduction from Bayes overlap to zero counts. The Poisson count approximation is a standard ingredient; the specified trajectory-testing curve and constrained entropy optimization are separate deductions, not claims of a new general Poisson approximation theorem.

Author-hosted primary text: https://dornsife.usc.edu/larry-goldstein/wp-content/uploads/sites/221/2023/06/pacs-1.pdf.

For a growing set of marked rows, the relevant input is the process statement of Theorem 2 on printed page 406, with the neighborhoods and quantities in (4)–(6) on page 405. Its bound is

\|\mathcal L(X)-\mathcal L(Y)\|\le2(2b_1+2b_2+b_3).

The paper’s total-variation norm is twice the supremum over events. Each coordinate of the comparison process is an independent Poisson variable with the original Bernoulli coordinate’s mean. Aggregating coordinates preserves an upper bound on total variation; this permits grouping time-and-type events into row counts.

In the compensated parity model, a marked set of size m has total one-edge probability m/n. The exact two-step reset proves independence from the joint outside-neighborhood information, giving b3 = 0; nearby marked departures supply b1 + b2 = O(s m²/n²). The resulting growing-row comparison is useful for the q true signal rows when lambda q²/n tends to zero. It does not approximate the whole ambient collection of background rows. Total variation alone also does not transfer expectations of dimension-dependent losses at the square-root-q scale; that step needs the separately established one-row relative estimate.