pith. sign in

arxiv: 1907.05942 · v1 · pith:KITV3LT3new · submitted 2019-07-12 · 🧮 math.CA · math.PR

The spectral matrices associated with the stochastic Darboux transformations of random walks on the integers

Manuel D. de la Iglesia , Claudia Juarez This is my paper

Pith reviewed 2026-05-24 21:53 UTC · model grok-4.3

classification 🧮 math.CA math.PR
keywords random walks on integersDarboux transformationsstochastic factorizationsspectral matricesGeronimus transformationsJacobi matricescontinued fractions
0
0 comments X

The pith

Spectral matrices for Darboux transformations of integer random walks are conjugations by degree-one matrix polynomials of Geronimus transformations of the original matrix.

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

The paper establishes conditions using continued fractions for the free parameters in UL and LU stochastic factorizations of the transition probability matrix of random walks on the integers. These factorizations allow the construction of new families of such walks by inverting the order of the factors, known as Darboux transformations. The spectral matrices linked to these new walks are shown to be obtained through conjugation by a degree-one matrix polynomial applied to a Geronimus transformation of the original spectral matrix. This identification is demonstrated for the case of constant transition probabilities with or without an attractive or repulsive force.

Core claim

The spectral matrices associated with these Darboux transformations (in both cases) are basically conjugations by a matrix polynomial of degree one of a Geronimus transformation of the original spectral matrix.

What carries the argument

Spectral matrix of the doubly infinite tridiagonal stochastic Jacobi matrix, transformed first by a Geronimus transformation and then conjugated by a degree-one matrix polynomial.

If this is right

  • Conditions on the free parameter ensure the factorizations are stochastic, allowing construction of new transition matrices.
  • Inverting the factors via Darboux transformations generates new families of random walks on the integers.
  • The spectral matrices for the new walks are explicitly related to the original via the Geronimus transformation and polynomial conjugation.
  • The method applies to random walks with constant transition probabilities, including those with attractive or repulsive forces.

Where Pith is reading between the lines

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

  • The continued-fraction conditions may allow parameterization of all possible stochastic Darboux transformations for these walks.
  • The explicit spectral-matrix relation could be used to compare recurrence or transience between original and transformed walks.
  • Similar factorization techniques might apply to birth-death processes on other graphs or with different step distributions.

Load-bearing premise

The free parameter of both factorizations can be chosen in terms of certain continued fractions such that this stochastic factorization is always possible.

What would settle it

For the constant-transition-probability random walk, compute the spectral matrix after the Darboux transformation and check whether it equals the predicted conjugation by a degree-one matrix polynomial of the Geronimus transform of the original spectral matrix.

read the original abstract

We consider UL and LU stochastic factorizations of the transition probability matrix of a random walk on the integers, which is a doubly infinite tridiagonal stochastic Jacobi matrix. We give conditions on the free parameter of both factorizations in terms of certain continued fractions such that this stochastic factorization is always possible. By inverting the order of the factors (also known as a Darboux transformation) we get new families of random walks on the integers. We identify the spectral matrices associated with these Darboux transformations (in both cases) which are basically conjugations by a matrix polynomial of degree one of a Geronimus transformation of the original spectral matrix. Finally, we apply our results to the random walk with constant transition probabilities with or without an attractive or repulsive force.

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 / 1 minor

Summary. The paper examines UL and LU stochastic factorizations of the transition probability matrix for random walks on the integers, represented as doubly infinite tridiagonal stochastic Jacobi matrices. It provides conditions on the free parameter using continued fractions to ensure the factorization remains stochastic. Through Darboux transformations (inverting the factors), new families of random walks are generated. The spectral matrices for these transformations are identified as conjugations by a degree-one matrix polynomial of a Geronimus transformation of the original spectral matrix. The results are applied to random walks with constant transition probabilities, including cases with attractive or repulsive forces.

Significance. If the central claims hold, the work advances the spectral theory of infinite Jacobi matrices and orthogonal polynomials by supplying explicit continued-fraction conditions for stochastic factorizations and an explicit relation between the spectral matrices of the original and Darboux-transformed walks. The concrete application to constant-probability walks supplies verifiable examples.

major comments (2)
  1. [Section stating the conditions on the free parameter (UL/LU factorizations)] The assertion that continued-fraction choices for the free parameter of the UL and LU factorizations always yield non-negative entries with unit row sums for arbitrary initial measures on Z is load-bearing for the subsequent Darboux construction and spectral identification. The manuscript must demonstrate that the resulting infinite family of inequalities is satisfied simultaneously at every site; convergence of the continued fraction alone does not automatically guarantee the sign and summation conditions everywhere.
  2. [Section on spectral-matrix identification] The identification that the new spectral matrices are conjugations by a degree-one matrix polynomial of a Geronimus transformation of the original spectral matrix relies on the stochastic factorization being valid; any gap in the verification of the continued-fraction conditions therefore affects the central claim.
minor comments (1)
  1. [Final application section] In the application to constant transition probabilities, state the explicit values of the probabilities and the force parameter used in the examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the continued-fraction conditions. We respond point by point below.

read point-by-point responses

  1. Referee: [Section stating the conditions on the free parameter (UL/LU factorizations)] The assertion that continued-fraction choices for the free parameter of the UL and LU factorizations always yield non-negative entries with unit row sums for arbitrary initial measures on Z is load-bearing for the subsequent Darboux construction and spectral identification. The manuscript must demonstrate that the resulting infinite family of inequalities is satisfied simultaneously at every site; convergence of the continued fraction alone does not automatically guarantee the sign and summation conditions everywhere.

    Authors: The referee correctly identifies a point that requires stronger emphasis. The continued fractions are defined recursively from the local transition probabilities so that each finite approximant satisfies non-negativity and unit row sums at the corresponding sites by construction. Convergence of the continued fraction then yields the infinite case. Nevertheless, the manuscript would benefit from an explicit lemma establishing that the infinite system of inequalities holds simultaneously at every site. We will insert such a lemma (with a short inductive argument on the approximants) in the revised version. revision: yes

  2. Referee: [Section on spectral-matrix identification] The identification that the new spectral matrices are conjugations by a degree-one matrix polynomial of a Geronimus transformation of the original spectral matrix relies on the stochastic factorization being valid; any gap in the verification of the continued-fraction conditions therefore affects the central claim.

    Authors: We agree that the spectral-matrix identification is conditional on the factorizations being stochastic. Once the additional lemma on the continued-fraction conditions is included, the argument in this section remains valid without further alteration. We will add a brief forward reference to the new lemma at the beginning of the spectral identification. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation constructs spectral identification from explicit continued-fraction conditions on the factorization parameter.

full rationale

The paper states conditions on the free parameter via continued fractions to guarantee stochastic UL/LU factorizations of the Jacobi matrix, then derives the associated spectral matrices as conjugations by a degree-1 polynomial of a Geronimus transform of the original matrix. This is a forward constructive chain (parameter choice enables factorization, which enables the Darboux step, which yields the spectral form) rather than any reduction of the claimed spectral identification to a fitted input, self-definition, or self-citation chain. No load-bearing step equates a result to its own inputs by construction, and the provided text contains no self-citations invoked as uniqueness theorems. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

The central claim rests on the existence of stochastic factorizations under continued-fraction conditions on a free parameter and on the algebraic relation between the transformed and original spectral matrices; no further axioms or invented entities are visible in the abstract.

free parameters (1)
  • free parameter of the UL and LU factorizations
    Mentioned explicitly as the quantity whose value must satisfy continued-fraction conditions for the factorization to remain stochastic.

pith-pipeline@v0.9.0 · 5658 in / 1116 out tokens · 22302 ms · 2026-05-24T21:53:47.928974+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Berezans’kii, Ju M., Expansions in Eigenfunctions of Selfadjoint Operators , Translations of Mathematical Monographs 17, American Mathematical Society, Rhode Island, 1968

    work page 1968

  2. [2]

    and W ang, X., Doubly infinite Jacobi matrices revisited: resolvent and sp ectral measure, Adv

    Dai, D., Ismail, M.E.H. and W ang, X., Doubly infinite Jacobi matrices revisited: resolvent and sp ectral measure, Adv. Math. 343 (2019), 157–192

    work page 2019

  3. [3]

    and Zygmunt, M., Matrix measures and random walks with a block tridiagonal tr ansition matrix, SIAM J

    Dette, H., Reuther, B., Studden, W. and Zygmunt, M., Matrix measures and random walks with a block tridiagonal tr ansition matrix, SIAM J. Matrix Anal. Applic. 29 (2006), 117–142

    work page 2006

  4. [4]

    and Yakimov, M., Noncommutative bispectral Darboux transformations , Trans

    Geiger, J., Horozov, E. and Yakimov, M., Noncommutative bispectral Darboux transformations , Trans. Amer. Math. Soc. 369 (2017), 5889–5919

    work page 2017

  5. [5]

    Grassmann, W.K., Means and variances of time averages in Markovian environme nts, Eur. J. Oper. Res. 31 (1987), 132–139

    work page 1987

  6. [6]

    Grassmann, W.K., Means and variances in Markov reward systems , in Linear Algebra, Markov Chains and Queueing Models, ed. C.D. Meyer and R.J. Plemmons. Springer-Verlag, NY, 1993

    work page 1993

  7. [7]

    Gr¨ unbaum, F.A.,Random walks and orthogonal polynomials: some challenges , Probability, Geometry and Integrable Systems, MSRI Publication, volumen 55, 2007

    work page 2007

  8. [8]

    Gr¨ unbaum, F.A., QBD processes and matrix orthogonal polynomials: some new e xplicit examples , Numerical Methods for Structured Markov Chains, eds. D. Bini, B. Meini, V. Ramaswa mi, M.A. Remiche and P. Taylor, Dagstuhl Seminar Proceeding s, 2008

    work page 2008

  9. [9]

    van den Ban and J.A.C

    Gr¨ unbaum, F.A., The Darboux process and a noncommutative bispectral proble m: some explorations and challenges , in E.P. van den Ban and J.A.C. Kolk (eds.), Geometric Aspects of Analysis and Mechanics: In Honor of the 65th Birthday of Hans Duistermaat, Progress in Mathematics 292, Springer, 2011

    work page 2011

  10. [10]

    and de la Iglesia, M.D., Matrix-valued orthogonal polynomials arising from group r epresentation theory and a family of quasi-birth-and-death processes , SIAM J

    Gr¨ unbaum, F.A. and de la Iglesia, M.D., Matrix-valued orthogonal polynomials arising from group r epresentation theory and a family of quasi-birth-and-death processes , SIAM J. Matrix Anal. Applic. 30 (2008), 741–761

    work page 2008

  11. [11]

    and de la Iglesia, M.D., Stochastic LU factorizations, Darboux transformations an d urn models , J

    Gr¨ unbaum, F.A. and de la Iglesia, M.D., Stochastic LU factorizations, Darboux transformations an d urn models , J. Appl. Prob. 55 (2018), 862–886

    work page 2018

  12. [12]

    and de la Iglesia, M.D., Stochastic Darboux transformations for quasi-birth-and- death processes and urn models, J

    Gr¨ unbaum, F.A. and de la Iglesia, M.D., Stochastic Darboux transformations for quasi-birth-and- death processes and urn models, J. Math. Anal. Appl. 478 (2019), 634–654

    work page 2019

  13. [13]

    and Haine, L., Orthogonal polynomials satisfying differential equations : the role of the Darboux transforma- tion, in: D

    Gr¨ unbaum, F.A. and Haine, L., Orthogonal polynomials satisfying differential equations : the role of the Darboux transforma- tion, in: D. Levi, L. Vinet, P. Winternitz (Eds.), Symmetries an I ntegrability of Differential Equations, CRM Proc. Lecture Notes, vol. 9, Amer. Math. Soc. Providence, RI, 1996, 143–15 4

    work page 1996

  14. [14]

    and Horozov, E., Some functions that generalize the Krall-Laguerre polynom ials, J

    Gr¨ unbaum, F.A., Haine, L. and Horozov, E., Some functions that generalize the Krall-Laguerre polynom ials, J. Comp. Appl. Math. 106 (1999), 271–297

    work page 1999

  15. [15]

    and Tirao, J.A., Matrix-valued spherical functions associated to the compl ex projective plane , J

    Gr¨ unbaum, F.A., Pacharoni, I. and Tirao, J.A., Matrix-valued spherical functions associated to the compl ex projective plane , J. Functional Analysis 188 (2002), 350–441

    work page 2002

  16. [16]

    and Tirao, J.A., A matrix-valued solution to Bochner’s problem , J

    Gr¨ unbaum, F.A., Pacharoni, I. and Tirao, J.A., A matrix-valued solution to Bochner’s problem , J. Physics A: Math. Gen. 34 (2001), 10647–10656

    work page 2001

  17. [17]

    and Tirao, J

    Gr¨ unbaum, F.A., Pacharoni, I. and Tirao, J. A., Two stochastic models of a random walk in the U( n)-spherical duals of U(n + 1), Ann. Mat. Pura Appl. 192 (2013), 447–473

    work page 2013

  18. [18]

    Heyman, D.P., A decomposition theorem for infinite stochastic matrices , J. Appl. Prob. 32 (1995), 893–903

    work page 1995

  19. [19]

    and Rom´ an, P., Some bivariate stochastic models arising from group repres entation theory , Stoch

    de la Iglesia, M.D. and Rom´ an, P., Some bivariate stochastic models arising from group repres entation theory , Stoch. Proc. Appl. 128 (2018), 3300–3326

    work page 2018

  20. [20]

    Nevai (editor), Kluwer Acad

    Ismail, M.E.H., Letessier, J., Masson, D., and Valent, G., Birth and death processes and orthogonal polynomials , in Orthogonal Polynomials, P. Nevai (editor), Kluwer Acad. Publishers (1 990), 229–255

    work page

  21. [21]

    and McGregor, J., The differential equations of birth and death processes, and the Stieltjes moment problem , Trans

    Karlin, S. and McGregor, J., The differential equations of birth and death processes, and the Stieltjes moment problem , Trans. Amer. Math. Soc., 85 (1957), 489–546

    work page 1957

  22. [22]

    and McGregor, J., The classification of birth-and-death processes , Trans

    Karlin, S. and McGregor, J., The classification of birth-and-death processes , Trans. Amer. Math. Soc., 86 (1957), 366–400

    work page 1957

  23. [23]

    and McGregor, J., Random walks , IIlinois J

    Karlin, S. and McGregor, J., Random walks , IIlinois J. Math., 3 (1959), 66–81

    work page 1959

  24. [24]

    and Ramaswami, V., Introduction to Matrix Analytic Methods in Stochastic Mode ling, ASA-SIAM Series on Statistics and Applied Probability, 1999

    Latouche, G. and Ramaswami, V., Introduction to Matrix Analytic Methods in Stochastic Mode ling, ASA-SIAM Series on Statistics and Applied Probability, 1999

    work page 1999

  25. [25]

    and Repka, J., Spectral theory of Jacobi matrices in ℓ2(Z) and the su(1,1) Lie algebra , SIAM J

    Masson, D.R. and Repka, J., Spectral theory of Jacobi matrices in ℓ2(Z) and the su(1,1) Lie algebra , SIAM J. Math. Anal., 22 (1991), 1131–1146

    work page 1991

  26. [26]

    and Salle, M.A., Differential-difference evolution equations II: Darboux tr ansformation for the Toda lattice , Lett

    Matveev, V.B. and Salle, M.A., Differential-difference evolution equations II: Darboux tr ansformation for the Toda lattice , Lett. Math. Phys. 3 (1979), 425–429

    work page 1979

  27. [27]

    Neuts, M.F., Structured Stochastic Matrices of M/G/ 1 Type and Their Applications , Marcel Dekker, New York, 1989

    work page 1989

  28. [28]

    Pruitt, W.E., Bilateral birth and death processes , Trans. Amer. Math. Soc. 107 (1962), 508–525

    work page 1962

  29. [29]

    and Zhedanov, A., Self-similarity, spectral transformations and orthogona l and biorthogonal polynomials in self-similar systems , V.B

    Spiridonov, V. and Zhedanov, A., Self-similarity, spectral transformations and orthogona l and biorthogonal polynomials in self-similar systems , V.B. Priezzhev and V.P.Spiridonov Editors. Proc. Interna tional W orkshop JINR. Dubna 1999. 349–361. 28 MANUEL D. DE LA IGLESIA AND CLAUDIA JUAREZ

    work page 1999

  30. [30]

    Vigon, V., LU factorization versus Wiener-Hopf factorization for Mar kov chains , Acta Appl. Math. 128 (2013), 1–37

    work page 2013

  31. [31]

    van Nostrand Co., N.Y., 1948

    W all, H.S., Analytic theory of continued fractions , D. van Nostrand Co., N.Y., 1948

    work page 1948

  32. [32]

    Korean Math

    Yoon, G.J., Darboux transforms and orthogonal polynomials , Bull. Korean Math. Soc. 39 (2002), 359–376

    work page 2002

  33. [33]

    Zhedanov, A., Rational spectral transformations and orthogonal polynom ials, J. of Comp. Appl. Math. 85 (1997), 67–86

    work page 1997

  34. [34]

    D 43 (1990), 269–287

    Zubelli, J.P., Differential equations in the spectral parameter for matrix differential operators, Phys. D 43 (1990), 269–287. Manuel D. de la Iglesia, Instituto de Matem ´aticas, Universidad Nacional Aut ´onoma de M ´exico, Circuito Exterior, C.U., 04510, Ciudad de M ´exico, M ´exico. E-mail address : mdi29@im.unam.mx Claudia Juarez, Instituto de Matem ´at...

    work page 1990