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arxiv: 2605.22146 · v1 · pith:QUXKE6KNnew · submitted 2026-05-21 · 🧮 math.PR

Poisson approximation of the largest gaps between zeros of a stationary Gaussian process

Renjie Feng , Stephen Muirhead This is my paper

Pith reviewed 2026-05-22 04:22 UTC · model grok-4.3

classification 🧮 math.PR
keywords stationary Gaussian processzeros of Gaussian processlargest gapsPoisson point processcorrelation decaysplitting propertyextreme value statistics
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The pith

If correlations decay at least polynomially, rescaled largest gaps between zeros of a stationary Gaussian process converge to a Poisson point process.

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

The paper shows that for a smooth stationary Gaussian process whose correlations decay at least polynomially, the largest gaps between successive zeros in a growing interval, after suitable rescaling of their locations and lengths, converge jointly to a Poisson point process. This holds because the authors prove an approximate splitting property with multiplicative error for gap events occurring in well-separated intervals. A reader would care since the result describes how the most extreme spacings behave like independent rare events, which controls the tail statistics of zero crossings in random fields. The novel technical step is extending the splitting argument to arbitrarily slow polynomial decay rates.

Core claim

For a smooth stationary Gaussian process with covariance decaying at least polynomially, the point process of rescaled locations and sizes of the largest gaps between zeros inside an interval of length T converges in distribution to a Poisson point process as T tends to infinity. The proof proceeds by establishing an approximate splitting property with multiplicative error for the probabilities of gap events in well-separated intervals, which remains valid even when the polynomial decay is arbitrarily slow.

What carries the argument

The approximate splitting property with multiplicative error for gap events in well-separated intervals, which upgrades the usual independence to a controlled multiplicative error and thereby yields the Poisson limit.

If this is right

  • Counts of gaps larger than any fixed threshold become approximately Poisson distributed and independent across distant subintervals.
  • The joint law of the k largest gaps and their positions inside the interval approaches the law of the k largest points of the limiting Poisson process.
  • The result applies uniformly to any polynomial decay rate, including rates slower than any fixed power.

Where Pith is reading between the lines

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

  • The same splitting technique may apply to other rare events for Gaussian processes, such as the locations of unusually large local maxima.
  • Numerical experiments could measure the rate at which the Poisson approximation improves as the interval length grows for processes with very slow correlation decay.
  • If local stationarity holds, the Poisson picture may extend to certain non-stationary Gaussian processes on the line.

Load-bearing premise

The Gaussian process is smooth and stationary, and its correlations decay at least polynomially fast.

What would settle it

Generate many independent realizations of a stationary smooth Gaussian process with polynomial correlation decay, locate the largest zero gaps inside successive large intervals of length T, rescale their positions and lengths, and test whether the empirical point pattern converges to a Poisson point process with the expected intensity.

read the original abstract

We study the largest gaps between successive zeros of a smooth stationary Gaussian process. Our main result is that, if correlations decay at least polynomially, then after suitable rescaling of the locations and sizes of the largest gaps in a growing interval, the resulting joint process converges to a Poisson point process. The main novel step in the proof is to establish an approximate splitting property, with multiplicative error, for gap events in well-separated intervals; notably we achieve this for processes with arbitrarily slow polynomial decay of correlations.

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 paper proves that for a smooth stationary Gaussian process whose covariance decays at least polynomially, the rescaled point process formed by the locations and lengths of the largest gaps between zeros inside a growing interval [0,T] converges in distribution to a Poisson point process with positive intensity. The central technical contribution is an approximate splitting (near-independence) property with multiplicative error for the gap events on well-separated subintervals, which is shown to hold even when the polynomial decay rate is arbitrarily slow.

Significance. If the limit theorem is established, the result supplies a Poisson approximation for extreme gap statistics under weak dependence, extending earlier work on Gaussian zeros to slower correlation decays. The manuscript supplies a novel approximate-splitting lemma with multiplicative error that is the load-bearing step; this is a genuine technical advance for the field.

major comments (2)
  1. [§3] §3 (Approximate splitting property, around the statement of the multiplicative-error bound): the error control for |r(t)| ≲ |t|^{-α} with small α>0 requires a minimal separation d(α) that grows at least like α^{-c} for some c>0. The manuscript must explicitly verify that d(α) can still be chosen o(T) (or at worst T^ε for small ε) uniformly in the number of intervals, so that the number of admissible disjoint subintervals inside [0,T] tends to infinity; otherwise the finite-dimensional distributions cannot converge to those of a PPP with positive intensity.
  2. [§5] §5 (Convergence of finite-dimensional distributions): the passage from the approximate splitting to the PPP limit uses a standard Chen-Stein or moment-comparison argument, but the intensity measure is only identified after the splitting error is shown to be o(1) uniformly over the growing number of intervals; this uniformity must be checked against the α-dependent separation.
minor comments (2)
  1. [§2] Notation for the rescaling of gap lengths (Eq. (2.7) or nearby) should be stated once and used consistently; the current definition mixes the normalizing constants for location and size.
  2. [Theorem 1.1] The statement of the main theorem should include an explicit remark that the intensity of the limiting PPP is positive and finite under the stated polynomial-decay assumption.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and detailed report. The comments help clarify the conditions under which the Poisson limit holds for arbitrarily slow polynomial decays. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (Approximate splitting property, around the statement of the multiplicative-error bound): the error control for |r(t)| ≲ |t|^{-α} with small α>0 requires a minimal separation d(α) that grows at least like α^{-c} for some c>0. The manuscript must explicitly verify that d(α) can still be chosen o(T) (or at worst T^ε for small ε) uniformly in the number of intervals, so that the number of admissible disjoint subintervals inside [0,T] tends to infinity; otherwise the finite-dimensional distributions cannot converge to those of a PPP with positive intensity.

    Authors: We appreciate this important clarification. In our construction, the separation parameter d depends only on the fixed decay exponent α > 0 of the given process. For any fixed α, d(α) is a constant independent of T. We choose the number of subintervals k(T) to satisfy k(T) → ∞ and k(T) · d(α) = o(T) as T → ∞, which is feasible since d(α) is fixed. The approximate splitting holds uniformly for all such well-separated collections. We will add a short paragraph in §3 explicitly stating this choice of k(T) and confirming that the number of intervals diverges, thereby ensuring the finite-dimensional distributions can converge to a non-degenerate PPP. This will be done in the revised manuscript. revision: yes

  2. Referee: [§5] §5 (Convergence of finite-dimensional distributions): the passage from the approximate splitting to the PPP limit uses a standard Chen-Stein or moment-comparison argument, but the intensity measure is only identified after the splitting error is shown to be o(1) uniformly over the growing number of intervals; this uniformity must be checked against the α-dependent separation.

    Authors: We agree that verifying the uniformity of the error is essential for the argument. The multiplicative error in the splitting lemma is controlled by a quantity that depends on α and d(α) but does not grow with the number of intervals k or with T (provided the intervals are separated by at least d(α)). In the proof of the finite-dimensional convergence in §5, we apply the lemma to k(T) intervals and show that the total variation or moment discrepancy is bounded by k times the per-interval error plus the approximation error from the local statistics. Since the per-interval error is o(1) uniformly in the location (by stationarity) and the choice of k(T) is such that the accumulated error vanishes, the limit is indeed the desired PPP. We will insert an additional lemma or remark in §5 that explicitly bounds the error uniformly in k under the α-dependent separation, referencing the construction in §3. This revision will be included. revision: yes

Circularity Check

0 steps flagged

No circularity: limit theorem derived from Gaussian process tail estimates and correlation decay bounds

full rationale

The paper establishes a Poisson point process limit for rescaled largest gaps via an approximate splitting property for gap events in separated intervals. This property is proved from the assumed polynomial decay of the covariance function using standard Gaussian tail bounds and integral estimates on the correlation kernel; the separation distance is chosen depending on the decay exponent but remains fixed for each fixed decay rate, allowing sufficiently many disjoint intervals for the finite-dimensional distributions to converge. No step reduces a claimed prediction to a fitted parameter, self-definition, or load-bearing self-citation; the derivation is self-contained against external probabilistic estimates and does not invoke prior results by the same authors as an unverified uniqueness theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of smooth stationary Gaussian processes and the polynomial decay assumption; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption The process is smooth and stationary.
    Explicitly stated in the abstract as the setting for the study of zero gaps.
  • domain assumption Correlations decay at least polynomially.
    The condition required for the main convergence result and the approximate splitting property.

pith-pipeline@v0.9.0 · 5600 in / 1269 out tokens · 65088 ms · 2026-05-22T04:22:46.589337+00:00 · methodology

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Reference graph

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