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General proof of a limit related to AR(k) model of Statistics

T0 review · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For any autoregressive order k, the averaged multidimensional sum of products of root powers converges to an explicit rational formula depending only on the absolute value of the sum of the shifts.

arxiv 1908.00428 v1 pith:XTRLKVCY submitted 2019-08-01 math.ST stat.TH

Jan Vrbik This is my paper · ORCID
classification math.STstat.TH
keywords generalmodelproofrelatedstatisticsarticlearxivautoregressive
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Autoregressive models describe sequences where each value depends on earlier values plus random noise. To estimate the model's parameters, statisticians need averages of products of observations. These averages reduce to complicated sums involving powers of the characteristic roots. Earlier work handled models of order 2, 3, and 4 one case at a time. This paper proves a formula that works for every order k. The main result says that a particular averaged sum, taken over all choices of k indices and weighted by products of root powers with absolute differences, converges to a closed-form expression. The expression is a sum over the roots: for each root, take the root raised to the total shift S plus k minus 1, and multiply by a product over the other roots of a rational factor involving their squares and differences. The proof counts how often each term appears, then uses a generating function and partial fractions to extract the needed coefficient. The proof assumes the roots are distinct; the paper says that repeated roots can be handled by taking limits, but it does not work out those formulas. No data or code are involved. The value is practical: it gives statisticians a general way to evaluate the limits needed for moment approximations in autoregressive models of any order.
Extended reading notes

Core claim

The paper's central assertion is Eq. (7): A = lim_{n→∞} (1/n) ∑_{i1,...,ik=1}^n λ1^{|i1-i2-s1|} ... λk^{|ik-i1-sk|} equals ∑_{j=1}^k λ_j^{S+k-1} ∏_{ℓ≠j} (1-λ_ℓ^2)/((λ_j-λ_ℓ)(1-λ_jλ_ℓ)), with S=|s1+...+sk|. The claim is that this identity holds for general k, with repeated roots obtained by a limiting procedure.

Load-bearing premise

The theorem as stated requires the roots λ_i to be pairwise distinct, since the displayed formula divides by (λ_j−λ_ℓ). The paper says in Section 3 that repeated roots are handled by taking limits and that the resulting expressions are finite, but it gives no derivation of those limits. If that extension is not valid, the 'general' formula does not cover all AR(k) models with multiple characteristic roots, even though the earlier summation (6) explicitly allows multiple roots.

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Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No constants are fitted to data; the λ_i are inputs of the AR(k) model. The proof rests on the stationarity assumption, distinctness of roots for the stated formula, and standard generating-function and algebra facts. The repeated-root extension is asserted rather than derived.

assumptions (4)
  • domain assumption All roots λ_i of the characteristic polynomial satisfy |λ_i| < 1 (stationarity condition).
    Invoked in Section 1 and used to ensure B_S converges absolutely and the Laurent series has a nonempty annulus of convergence.
  • domain assumption For the main theorem, the λ_i are pairwise distinct; repeated roots are handled by a separate limiting argument in Section 3.
    Partial-fraction expansion with simple poles and denominators λ_j−λ_ℓ require distinct roots; Section 3 only sketches the repeated-root case.
  • standard math Standard facts about absolutely convergent series, term-by-term limits, and partial fraction expansions of rational functions.
    Used throughout Section 2, especially in the A≥B∞ argument and coefficient extraction from F(t).
  • standard math The Laurent series for F(t) is valid in the annulus max_ℓ |λ_ℓ| < |t| < min_ℓ |λ_ℓ|^(-1).
    Required for expanding each factor and reading off the coefficient of t^S.

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Pith. "Pith review of General proof of a limit related to AR(k) model of Statistics." pith.science (2026). https://pith.science/paper/XTRLKVCY

@misc{pith2026190800428,
  author       = {Pith},
  title        = {Pith review of: General proof of a limit related to AR(k) model of Statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XTRLKVCY}},
  note         = {Machine review of arXiv:1908.00428}
}
read the original abstract

Computing moments of various parameter estimators related to an autoregressive model of Statistics, one needs to evaluate several non-trivial limits. This was done by arXiv:1506.03131 for the case of two, three and four dimensions; in this article, we present a proof of a fully general formula, based on an ingenious solution of https://mathoverflow.net/users/4312/fedor-petrov.

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

  1. [1]

    Fedor Petrov (https://mathoverflow.net/users/4312/fedo r-petrov), Prove an existing formula for a limit of a specific sum, URL (version: 2019-07-0 9): https://mathoverflow.net/q/335816

    work page 2019

  2. [2]

    150504 (May 4, 2015)

    Yuhao Liu: ”Finding moments of AR(k)-model parameter estimat ors” Brock Reports in Mathematics and Statistics No. 150504 (May 4, 2015)

    work page 2015

  3. [3]

    Yuhao Liu and Jan Vrbik: https://arxiv.org/abs/1506.03131

    work page Pith review arXiv

  4. [4]

    Jan Vrbik: ”Moments of AR(k) parameter estimators” Communications in Statistics - Simulation and Computation 44 (2015) 1239-1252 5

    work page 2015

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Reviewed August 14, 2026 · model on record in the stance chip above.