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Moments of balanced P\'olya urns

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Pith's one-line read For balanced Polya urns, the p-th moments of centered ball counts obey explicit polynomial-in-n bounds matching the urn's dominant eigenvalues, yielding moment convergence whenever a central limit theorem is known.

desk verdict Janson proves clean Lp moment bounds for balanced Polya urns without irreducibility; the proof is checkable and the limitations are honestly stated. read the letter →

arxiv 2505.14198 v1 pith:DF5OFSSB submitted 2025-05-20 math.PR

Svante Janson This is my paper · ORCID
classification math.PR
keywords resultsbalancedboundsknownmomentsolyaurnsasymptotic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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The reading

Imagine an urn with balls of several colors. At each step you draw one ball at random and, depending on its color, add or remove a random collection of balls. This is a Polya urn. The paper studies balanced urns, where the total added weight is the same fixed number after every draw, so the urn's total weight grows linearly and is known exactly at each step.

The main objects are the centered counts X_n - E X_n: how far the number of balls of each color is from its mean at time n. The paper bounds the p-th moment, a measure of spread averaged and raised to the power p, by an explicit power of n, possibly with a logarithmic correction. The power depends only on the two largest eigenvalues of the urn's intensity matrix. When the urn is small in the spectral sense, the bound is n^{1/2}; when it is large, a larger power of n appears. These bounds match the normalizations used in known central limit theorems.

The method is general and relatively simple. The increments are split into a martingale difference sequence plus a deterministic linear term. Burkholder's inequality controls sums of martingale differences, and estimates for products of matrices of the form I + A/w_k translate the spectral data of the intensity matrix into the right growth rates. Because the Lp norms are bounded by the correct normalizing factors, any known convergence in distribution automatically upgrades to convergence of all moments. The paper extends earlier moment estimates by the same author and Pouyanne, but it removes the irreducibility assumption and allows random replacements.

Extended reading notes

Core claim

The load-bearing result is Theorem 3.1: for a tenable balanced urn with λ1 = b and finite p-th replacement moments, ||X_n - E X_n||_p ≤ C_p n^{1/2} when Re λ2 < λ1/2, ≤ C_p n^{1/2} (log n)^{ν2+1/2} when Re λ2 = λ1/2, and ≤ C_p n^{Re λ2/λ1} (log n)^{ν2} when Re λ2 > λ1/2. Theorems 3.2 and A.1 convert this into convergence of all moments (up to order p) whenever the corresponding distributional limit is known.

Load-bearing premise

The argument collapses without the balance condition (PU4), a · ξ_i = b a.s. for every color, because it is what makes the total weight w_n = w_0 + nb deterministic and gives the non-random matrix product representation (4.12)-(4.14). If balance is dropped, the weight sequence becomes random, the recurrence (4.11) no longer has deterministic coefficients, and the paper notes (Problem 1.1 and [14, Example 14.2]) that moment convergence can fail in unbalanced examples. All theorems in the paper are conditional on this structural assumption.

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Summary. This paper studies balanced generalized Pólya urns in discrete time. Theorem 3.1 gives, under tenability, balance, and λ1=b, upper bounds for ||X_n - E X_n||_p for p≥2 in three regimes determined by Re λ2: C_p n^{1/2}, C_p n^{1/2}(log n)^{ν2+1/2}, or C_p n^{Re λ2/λ1}(log n)^{ν2}. Theorem 3.3 refines this to individual spectral projections Pλ(X_n - E X_n), and Theorem 3.4 shows that the λ1 component has zero centered fluctuation when λ1 is simple. Theorem 3.2 converts the bounds into convergence of all moments under previously known asymptotic normality results; Appendix A provides a uniform-integrability version when only a single p≥2 is assumed. The proofs use a martingale difference decomposition Y_n, the balance condition to obtain a deterministic recurrence and matrix-product representation F_{ℓ,n}, Burkholder-type inequalities (Lemma 5.1), and spectral estimates for those matrix products (Lemmas 6.1 and 6.2).

Significance. If the result holds, it is a useful unification: it recovers and extends earlier moment bounds for balanced urns with a simpler proof, and it upgrades known central limit theorems to moment convergence in the small-urn and critical regimes. The proof is checkable and essentially self-contained, with complete proofs included for the key lemmas. All constants are unspecified but not fitted, and the load-bearing structural assumption (balance, PU4) is explicit; the paper also honestly states the open problem of unbalanced urns. The theorems assume λ1=b directly, so the proof does not depend on the correctness of the cited Lemma 2.2. Overall this is a solid, useful contribution to the Pólya urn literature.

minor comments (5)
  1. [Section 4, proof of Theorem 3.2] The step concluding uniform integrability is too terse: Theorem 3.3 is stated for p≥2, and boundedness of L^r norms for all r would be needed to upgrade convergence in distribution to moment convergence of order p. The argument works by applying Theorem 3.3 at an exponent q>p, but this should be stated explicitly.
  2. [Appendix A, Theorem A.1] Theorem A.1 invokes Lemma A.2 without verifying its hypothesis that |Y_i|^p is uniformly integrable. The verification is short: under (3.1), the conditional law of ΔX_n is a mixture of the finitely many fixed laws of ξ_j, each with finite p-th moment, and the conditional expectation term in Y_i is a.s. bounded by C^{1/p}; adding this one-line check would make the appendix self-contained.
  3. [Section 2.1 and (4.1)] The notation ΔX_n is overloaded: it denotes both the generic replacement vector in the paragraph after (2.1) and the actual increment X_{n+1}-X_n in (4.1). Using different symbols for the generic replacement and the realized increment would avoid confusion.
  4. [Lemma 6.1] The passage from the case i≥i0 to all i≥1 is asserted in a single sentence. The finite number of omitted factors are indeed bounded, but spelling out this boundedness would make the proof fully explicit.
  5. [Section 1 and Remark 3.5] There are small typos: 'irrdeucible' should be 'irreducible' in Section 1, and 'strictly less that p' in Remark 3.5 should be 'strictly less than p'.
Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper is a proof-based contribution with no data fitting. The assumptions are explicit domain conditions (balance, tenability, finite moments) and standard mathematics (Burkholder, Jordan decomposition). The only imported Polya-specific result is Lemma 2.2 from the author's [13], used to justify λ1 = b.

assumptions (6)
  • domain assumption Tenability (2.2): every X_ni ≥ 0 and Σ_i a_i X_ni > 0 a.s. at all times.
    Stated in Section 2.1 as a standing assumption; without it the drawing probabilities in (2.1) are not well defined and the process can stop.
  • domain assumption Balance (PU4): Σ_j a_j ξ_ij = b > 0 a.s. for every colour i.
    The central restriction of the paper; makes w_n = w_0 + nb deterministic and enables the linear recurrence (4.11) with non-random matrix products.
  • domain assumption Finite replacement moments: E|ξ_ij|^p < ∞ for the needed orders (3.1).
    Used to get ||ΔX_n||_p ≤ C_p in (4.8)-(4.10); Theorem 3.2 assumes this for every p ≥ 1, and Appendix A relaxes it to a single p.
  • domain assumption λ1 = b, meaning the balance eigenvalue is the rightmost; justified by Lemma 2.2 from [13].
    Assumed in Section 3 for all theorems and used to label the dominant eigenvalue in Theorems 3.1 and 3.3; the cited lemma gives conditions under which it holds.
  • standard math Burkholder's square-function inequality for martingales, scalar and vector-valued.
    Used in Lemma 5.1 and Appendix A; standard, see [7] and [10].
  • standard math Jordan normal form and spectral projections satisfying (2.5)-(2.6).
    Used throughout Sections 2-6; the projection P_λ and nilpotent N_λ encode the eigenstructure needed for the matrix estimates.

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Pith. "Pith review of Moments of balanced P\'olya urns." pith.science (2026). https://pith.science/paper/DF5OFSSB

@misc{pith2026250514198,
  author       = {Pith},
  title        = {Pith review of: Moments of balanced P\'olya urns},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DF5OFSSB}},
  note         = {Machine review of arXiv:2505.14198}
}
read the original abstract

We give bounds for (central) moments for balanced P\'olya urns under very general conditions. In some cases, these bounds imply that moment convergence holds in earlier known results on asymptotic distribution. The results overlap with previously known results, but are here given more generally and with a simpler proof.

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Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [1]

    Athreya and Samuel Karlin: Embedding of urn schemes into continuous time Markov branching processes and related limit the- orems

    Krishna B. Athreya and Samuel Karlin: Embedding of urn schemes into continuous time Markov branching processes and related limit the- orems. Ann. Math. Statist. 39 (1968), 1801–1817

    work page 1968

  2. [2]

    Bagchi and A

    A. Bagchi and A. K. Pal: Asymptotic normality in the generalized P´ olya–Eggenberger urn model, with an application to computer data structures. SIAM J. Algebraic Discrete Methods 6 (1985), no. 3, 394– 405

    work page 1985

  3. [3]

    Stochastic Process

    Zhi-Dong Bai and Feifang Hu: Asymptotic theorems for urn models with nonhomogeneous generating matrices. Stochastic Process. Appl. 80 (1999), no. 1, 87–101

    work page 1999

  4. [4]

    Zhi-Dong Bai and Feifang Hu: Asymptotics in randomized urn models. Ann. Appl. Probab. 15 (2005), no. 1B, 914–940

    work page 2005

  5. [5]

    Bernstein: Nouvelles applications des grandeurs al´ eatoires presqu’ind´ ependantes

    Sergei N. Bernstein: Nouvelles applications des grandeurs al´ eatoires presqu’ind´ ependantes. (Russian)Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), no. 2, 137–150

    work page 1940

  6. [6]

    Bernstein: Sur un probl` eme du sch´ ema des urnes ` a compo- sition variable

    Sergei N. Bernstein: Sur un probl` eme du sch´ ema des urnes ` a compo- sition variable. C. R. (Doklady) Acad. Sci. URSS (N.S.) 28 (1940), 5–7

    work page 1940

  7. [7]

    Burkholder: Martingale transforms

    Donald L. Burkholder: Martingale transforms. Ann. Math. Statist. 37 (1966), 1494–1504. 16 SV ANTE JANSON

    work page 1966

  8. [8]

    Burkholder: Explorations in martingale theory and its ap- plications

    Donald L. Burkholder: Explorations in martingale theory and its ap- plications. ´Ecole d’ ´Et´ e de Probabilit´ es de Saint-Flour XIX—1989, 1–66. Lecture Notes in Math. 1464, Springer-Verlag, Berlin, 1991

    work page 1989

Show all 18 references
  1. [9]

    Eggenberger and G

    F. Eggenberger and G. P´ olya:¨Uber die Statistik verketteter Vorg¨ ange. Zeitschrift Angew. Math. Mech. 3 (1923), no. 4, 279–289

    1923

  2. [10]

    Allan Gut: Probability: A Graduate Course , 2nd ed., Springer, New York, 2013

    2013

  3. [11]

    Svante Janson: Functional limit theorems for multitype branching pro- cesses and generalized P´ olya urns. Stoch. Process. Appl. 110 (2004), 177–245

    2004

  4. [12]

    Svante Janson: Limit theorems for triangular urn schemes. Probab. Theory Rel. Fields 134 (2005), 417–452

    2005

  5. [13]

    Svante Janson: Mean and variance of balanced P´ olya urns. Adv. in Appl. Probab. 52:4 (2020), 1224–1248

    2020

  6. [14]

    Preprint, 2024.arXiv:2402.01299

    Svante Janson: Almost sure and moment convergence for triangular P´ olya urns. Preprint, 2024.arXiv:2402.01299

    2024 arXiv

  7. [15]

    Svante Janson and Nicholas Pouyanne: Moment convergence of bal- anced P´ olya processes.Electron. J. Probab. 23 (2018), Paper No. 34, 13 pp

    2018

  8. [16]

    Mahmoud: P´ olya Urn Models

    Hosam M. Mahmoud: P´ olya Urn Models. CRC Press, Boca Raton, FL, 2009

    2009

  9. [17]

    Markov: Sur quelques formules limites du calcul des prob- abilit´ es

    Andrey A. Markov: Sur quelques formules limites du calcul des prob- abilit´ es. (Russian.)Bulletin de l’Acad´ emie Imp´ eriale des Sciences11 (1917), no. 3, 177–186

    1917

  10. [18]

    Nicolas Pouyanne: An algebraic approach to P´ olya processes.Ann. Inst. Henri Poincar´ e Probab. Stat.44 (2008), no. 2, 293–323. Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden Email address : svante.janson@math.uu.se URL: http://www.math.u...

    2008

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