bibkey: “siripraparat2021local” authors: “Tatpon Siripraparat; Kritsana Neammanee” year: 2021 title: “A local limit theorem for Poisson binomial random variables” doi: “10.2306/scienceasia1513-1874.2021.006” url: “https://www.scienceasia.org/2021.47.n1/scias47_111.pdf” claim: “Theorem 2 bounds the uniform point-probability error for a heterogeneous independent Bernoulli sum by an explicit quantity of order inverse total variance.” strata_touched: [] license: “citation-only” triage: “anchor”

Local Bernoulli probabilities and a growing posterior interval

Published in ScienceAsia 47 (2021), 111–116. The checked primary publisher PDF has six pages. The introduction on printed page 111 defines the uniform point-probability error against the normal density divided by its standard deviation. Theorem 2 on printed page 112 applies to independent Bernoulli variables with heterogeneous parameters and total variance greater than one. Its explicit bound is of order inverse total variance as that variance grows. It requires neither identical parameters nor uniform separation from zero and one. This local bound is literature-attested; an ordinary distribution- function Berry–Esseen bound alone does not supply the needed relative central point probabilities.

The parity fluctuation volume, Chapter 32 uses this result for a calibrated independent Bernoulli representation of the fixed-size posterior, whose total mean equals the support size exactly. It applies the theorem to both the full auxiliary array and the complement of the data-selected rank interval. The resulting ratio is integrated under the interval’s product law. This controls its full posterior label vector in total variation, then its exact conditional mean at the smaller fluctuation scale. The source’s independent Bernoulli variables are auxiliary posterior variables; they are not the actual dependent observations.

The local theorem does not itself establish the actual rank interval’s size, its confinement to adjacent lattice layers, the calibrated posterior variance, or its coupling to the coarse loss coordinate. Those steps and their joint mixed-normal conclusion in the stationary pair/path experiment are repo-derived ordinary mathematics. Generic conditioning approximations and Gaussian variance mixtures are classical mechanisms, not new general theories. The posterior conditional statement is under the uniform support prior; permutation equivariance transfers the unconditional joint law to every fixed support. No conditional random-label claim is made with the true support fixed.

The total-variation estimate uses an interval carrying a vanishing fraction of total auxiliary variance. A fixed positive fraction retains a finite-population variance correction. The result therefore does not claim that conditioning can be removed from arbitrary large subsets, nor that all posterior coordinates are jointly independent. It also does not assert convergence of actual fourth moments or make a global originality claim.

Selected posterior noise on two simultaneous scales

Chapter 35 of the same fluctuation volume uses Theorem 2 for an entire selected set together with both boundary rank prefixes. The selected set has order-support-size cardinality, while its auxiliary Bernoulli variance is a vanishing fraction of the full variance. The local theorem is applied to the full sum and its complement, both with variance comparable to the support size. Its uniform absolute point error therefore yields the needed density ratio for every selected count. Small cardinality is not the premise used by this calculation.

Transferring the center requires more than multiplying total variation by the size of the selected set. The proof integrates a centered subset sum against the conditioning ratio; the constant term cancels. Cauchy–Schwarz and the independent Bernoulli fourth-moment bound then give one variance-weighted bound for every subset of the same union. It controls the selected total and all boundary prefixes at their different normalizations. These are direct probability estimates inside the model-specific proof, not claims of a new general local limit theorem.

The selected variance constant comes from exact signal/background change of measure and a logistic-weighted local integral, or its lattice sum beginning at the first accepted layer. The fixed-width nonlattice input is attributed to Stone. The independent-array process limit and the conditional-kernel independence mechanism are attributed to Whitt and Pasquazzi. The proof verifies joint Gaussian convergence before using the vanishing selected/boundary overlap covariance to conclude independence.

Arratia–Goldstein–Langholz (2005) provides the classical conditional Bernoulli representation in Lemma 3.5. Its stronger local expansions use Condition 2.1, including a lower bound on variance proportional to the number of coordinates. That condition is not assumed for this full sparse array. The total-variance local bound above suffices, so no sharper inclusion-probability expansion is imported.

The resulting joint actual pair/path law with exact posterior centers, a constant intermediate-noise curve and an independent fine boundary process is repo-derived ordinary mathematics. The auxiliary local theorem, rejective sampling, weak-convergence methods and Gaussian independence criterion are literature-attested. The inspected sources do not supply the entire actual-model bridge. This is a bounded source comparison, not a global originality certificate; the statement asserts weak convergence, not convergence of actual moments or finite-sample posterior independence.

Microscopic windows and uniform anti-concentration

The posterior threshold volume, Chapter 37 uses the same local theorem in two roles. Applying it to the calibrated whole sum and the complementary sum removes the fixed-size conditioning on a window with a vanishing fraction of total auxiliary variance. Applying it to the window sum itself bounds every point probability by an absolute constant divided by its standard deviation. Counting the integers in a real interval gives a bound uniform in the interval’s center. The exact posterior center can therefore be used directly, without transferring an unbounded mean through total variation. These local-probability and anti-concentration arguments are classical.

The model-specific input is an actual central count group in a nonlattice score law. Its signal probability follows from Stirling’s formula; exact untilting supplies a background contribution of the same leading order. The one- and two-row comparisons give concentration of the mixed group in both stationary experiments. Its auxiliary variance is of order q/lambda, whereas containment in a fixed-width score interval gives an o(q) upper bound for the entire shrinking window. The argument needs no relative shrinking-window density estimate and no exact asymptotic for the whole window variance.

The resulting escape from every bounded interval at normalization q^(1/4), with exact posterior centering, is repo-derived ordinary mathematics. The elementary endpoint-continuity argument then excludes J1 tightness of the corresponding microscopic score-window process. This strengthens Chapter 33’s single-row density obstruction. It does not contradict the capacity-indexed process limits or the macroscopic threshold-field limit, and it asserts no universal nonlattice tangent law or convergence of actual moments. The attribution of the auxiliary tools is not a global originality certificate for the model conclusion.