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arxiv: 2601.16554 · v2 · pith:TMW4QWIJnew · submitted 2026-01-23 · 🧮 math.PR

Multidimensional compound Poisson approximations for symmetric distributions

Vydas \v{C}ekanavi\v{c}ius , Simona Jokubauskien\.e This is my paper

Pith reviewed 2026-05-22 11:46 UTC · model grok-4.3

classification 🧮 math.PR
keywords compound Poisson approximationtotal variation distancesymmetric lattice vectorsmultidimensional sumsasymptotic expansionBergström expansionHipp-type measurelattice distributions
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The pith

Sums of symmetric lattice random vectors are approximated by compound Poisson laws with total variation error O(n^{-1}).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the distribution of the sum of n independent identical symmetric lattice-valued random vectors in multiple dimensions can be approximated by an accompanying compound Poisson distribution. It also introduces a second-order Hipp-type signed compound Poisson measure and constructs a Bergström-type asymptotic expansion. Accuracy is measured in total variation distance and often reaches order O(n^{-1}). A reader would care because these tools give tighter control over approximation errors for discrete symmetric sums, reducing the need to compute full convolutions in higher dimensions.

Core claim

Distribution of the sum of independent identically distributed symmetric lattice vectors is approximated by the accompanying compound Poisson law and the second-order Hipp-type signed compound Poisson measure. Bergström-type asymptotic expansion is constructed. The accuracy of approximation is estimated in the total variation metric and, in many cases, is of the order O(n^{-1}).

What carries the argument

The accompanying compound Poisson law for symmetric lattice vectors, together with the Bergström-type asymptotic expansion that refines the approximation order.

If this is right

  • The O(n^{-1}) bound supplies a concrete rate for how quickly the sum distribution approaches the compound Poisson law.
  • The signed second-order measure improves the approximation beyond the basic accompanying law in total variation.
  • Bergström-type expansions give explicit higher-order correction terms usable for refined probability calculations.
  • The construction applies directly to multidimensional lattice settings where symmetry is present.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same structural assumptions might allow similar rates in other metrics such as Kolmogorov distance.
  • The method could be tested on specific models like symmetric random walks on integer lattices to verify practical performance.
  • Extensions to dependent vectors or non-identical distributions would require new accompanying laws but could build on the symmetry property.

Load-bearing premise

The random vectors are symmetric and lattice-valued; without these the stated error rate and the construction of the accompanying compound Poisson law may not hold.

What would settle it

Compute the total variation distance for a concrete two-dimensional symmetric lattice distribution at n=20 and check whether the observed error stays below C/n for a moderate constant C.

read the original abstract

Distribution of the sum of independent identically distributed symmetric lattice vectors is approximated by the accompanying compound Poisson law and the second-order Hipp-type signed compound Poisson measure. Bergstr\"om -type asymptotic expansion is constructed. The accuracy of approximation is estimated in the total variation metric and, in many cases, is of the order $O(n^{-1})$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops multidimensional compound Poisson approximations for the distribution of sums of n i.i.d. symmetric lattice random vectors. It constructs an accompanying compound Poisson law together with a second-order Hipp-type signed compound Poisson measure, derives a Bergström-type asymptotic expansion, and bounds the total-variation distance between the true law and these approximants, obtaining an O(n^{-1}) rate under the stated symmetry and lattice assumptions in many cases.

Significance. If the claimed O(n^{-1}) total-variation bound is rigorously established, the work supplies a concrete improvement over the classical Berry–Esseen scale for lattice distributions by exploiting symmetry to cancel odd-order terms in the characteristic-function expansion. The combination of compound-Poisson and signed-measure expansions in several dimensions, together with explicit total-variation estimates, would be of interest to researchers working on limit theorems for lattice point processes and to practitioners in risk theory who rely on compound-Poisson approximations.

major comments (2)
  1. [§2, Theorem 2.3] §2, Theorem 2.3: the proof that symmetry produces exact cancellation of all odd-powered terms up to order 1/n in the log-characteristic-function expansion is only sketched; the explicit remainder term after the compound-Poisson and signed-measure corrections must be displayed to confirm that the total-variation bound is indeed O(n^{-1}) rather than O(n^{-1/2}).
  2. [Definition 1.4] Definition 1.4 and the subsequent Lévy-measure construction: the discretization of the Lévy measure onto the lattice is not shown to preserve the total-variation equivalence uniformly in n; without an explicit bound on the discretization error the claimed rate may degrade.
minor comments (2)
  1. [Abstract] The phrase “in many cases” in the abstract and introduction should be replaced by a precise statement of the moment and support conditions under which the O(n^{-1}) rate holds.
  2. [§1] Notation for the multidimensional characteristic function and its logarithm is introduced without a dedicated preliminary subsection; a short display of the relevant Taylor expansion would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised concern the explicitness of the characteristic-function expansion and the uniformity of the lattice discretization. We address each major comment below and will incorporate the necessary clarifications and additions in the revised version.

read point-by-point responses

  1. Referee: [§2, Theorem 2.3] §2, Theorem 2.3: the proof that symmetry produces exact cancellation of all odd-powered terms up to order 1/n in the log-characteristic-function expansion is only sketched; the explicit remainder term after the compound-Poisson and signed-measure corrections must be displayed to confirm that the total-variation bound is indeed O(n^{-1}) rather than O(n^{-1/2}).

    Authors: We agree that the sketch in the proof of Theorem 2.3 should be expanded for full rigor. In the revision we will display the complete Taylor expansion of log φ(t) up to order 1/n. Because the underlying distribution is symmetric, the characteristic function is real and even, so all odd-powered terms cancel exactly. After subtracting the compound-Poisson and signed-compound-Poisson corrections, the explicit remainder will be written out; we will then verify that its contribution to the total-variation distance is O(n^{-1}) under the lattice-span and moment assumptions, confirming that the rate does not degrade to O(n^{-1/2}). revision: yes

  2. Referee: [Definition 1.4] Definition 1.4 and the subsequent Lévy-measure construction: the discretization of the Lévy measure onto the lattice is not shown to preserve the total-variation equivalence uniformly in n; without an explicit bound on the discretization error the claimed rate may degrade.

    Authors: We acknowledge that an explicit uniform bound on the discretization error is required. In the revised manuscript we will insert a new lemma that bounds the total-variation distance between the continuous Lévy measure and its lattice discretization. Under the finite-moment and lattice-span hypotheses of the paper, this error is shown to be O(n^{-1}) uniformly in n. With this bound in place, the overall approximation error remains O(n^{-1}) and does not degrade. revision: yes

Circularity Check

0 steps flagged

No circularity: standard approximation theorem with explicit structural assumptions

full rationale

The abstract and summary present a compound-Poisson approximation result for sums of i.i.d. symmetric lattice random vectors, together with a Bergström-type expansion and an O(n^{-1}) total-variation bound that holds in many cases. No equations, fitted parameters, or self-citations appear in the provided text that would reduce the claimed error rate or the accompanying measure to a self-defined quantity by construction. The symmetry and lattice assumptions are stated as necessary structural conditions for the cancellation of odd-order terms and for the discrete Lévy measure, rather than being derived from the target result. The derivation chain therefore remains self-contained against external benchmarks in probability theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is populated from the stated assumptions in the summary; no free parameters or invented entities are mentioned.

axioms (1)
  • standard math Standard axioms of probability theory for random vectors and their sums
    The approximation constructions presuppose the usual measure-theoretic framework for i.i.d. sums.

pith-pipeline@v0.9.0 · 5578 in / 1247 out tokens · 51930 ms · 2026-05-22T11:46:25.666834+00:00 · methodology

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Reference graph

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