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An MA(1) process with uniform innovations conditioned to stay positive converges to an explicit Doob h-transform in the non-expanding regime.

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2026-06-28 13:00 UTC pith:YV5RM4TG

load-bearing objection The paper works out explicit formulas for the eigenfunction h, persistence exponent, and phase-dependent limiting kernel of the conditioned MA(1) with uniform innovations when θ ∈ [-1,1), using the Markov representation and h-transform. the 2 major comments →

arxiv 2606.01832 v1 pith:YV5RM4TG submitted 2026-06-01 math.PR

MA(1) processes with uniform innovations conditioned to stay positive in the non-expanding regime

Frank Aurzada , Virginia Worf This is my paper
classification math.PR
keywords MA(1) processuniform innovationsconditioned to stay positiveDoob h-transformpersistence asymptoticsMarkov chainnon-expanding regimeeigenfunction
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that an MA(1) process driven by uniform innovations, when conditioned to remain positive for all times up to n, possesses limiting finite-dimensional distributions as n tends to infinity. Representing the process as a Markov chain on a continuous state space allows the authors to identify this limit as a Doob h-transform of the original chain. For the non-expanding parameter range θ ∈ [-1, 1), they derive explicit generating functions that produce closed-form expressions for the eigenfunction h, the persistence exponent, and the full transition kernel of the limit. The resulting object has a phase-dependent structure that changes with the value of θ. This supplies a concrete, solvable instance of persistence conditioning for a continuous-state Markov chain.

Core claim

For the MA(1) process with uniform innovations and coupling parameter θ ∈ [-1, 1), the process conditioned to stay positive up to time n converges in finite-dimensional distributions to the Doob h-transform of the underlying Markov chain; the eigenfunction h and the persistence exponent are obtained from explicit generating functions of the transition kernel, yielding a fully explicit transition kernel for the limiting process that depends on the phase of θ.

What carries the argument

The Doob h-transform of the Markov chain representation of the MA(1) process, where the eigenfunction h serves as the harmonic function that reweights paths according to their persistence probability.

Load-bearing premise

The MA(1) process with uniform innovations admits an exact representation as a Markov chain on a continuous state space, allowing standard Doob h-transform theory to identify the conditioned limit.

What would settle it

Numerical simulation of the conditioned MA(1) trajectories for a fixed θ in [-1, 1) would fail to reproduce the transition probabilities predicted by the explicit kernel derived from the generating functions.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The transition kernel of the limiting conditioned process is fully explicit and exhibits a phase-dependent structure in θ.
  • Sharp asymptotics for the probability of staying positive are available in closed form.
  • Explicit formulas exist for both the eigenfunction h and the persistence exponent.
  • The same representation yields the complete finite-dimensional distributions of the limit.

Where Pith is reading between the lines

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

  • The phase-dependent structure may indicate analogous transitions in related linear processes with different innovation laws.
  • The explicit kernel supplies a benchmark for testing approximation methods on conditioned continuous-state chains that lack closed forms.
  • Extensions to higher-order moving-average processes could be attempted if analogous Markov representations can be found.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The paper studies an MA(1) process with uniform innovations conditioned to remain positive. It represents the process as a Markov chain on a continuous state space, proves existence of the limiting finite-dimensional distributions under the conditioning, and identifies the limit as a Doob h-transform. For the non-expanding regime θ ∈ [-1,1), the authors compute the relevant generating functions, derive sharp persistence asymptotics, and obtain explicit formulas for the eigenfunction h and the persistence exponent, yielding a fully explicit transition kernel for the limiting process that depends on the phase of the parameters. This is presented as a rare solvable example of a continuous-state Markov chain conditioned on persistence.

Significance. If the central claims hold, the work supplies one of the few fully explicit examples of an h-transformed continuous-state chain arising from a linear process with uniform noise. The explicit generating functions, eigenfunction, and kernel would enable direct computation of stationary measures and transition probabilities in the conditioned regime, which is uncommon for non-lattice continuous-state models.

major comments (2)
  1. [Markov representation and h-transform sections] The load-bearing step is the claim that the augmented chain (X_n, ε_n) satisfies the irreducibility, positivity, and regularity conditions needed for standard Doob h-transform theory on a continuous state space to apply directly and produce a positive eigenfunction h with simple principal eigenvalue. The abstract asserts that the representation 'admits an exact Markov representation' to which the theory applies, but the skeptic correctly notes that these conditions are not automatic for MA(1) representations; the manuscript must contain an explicit verification (e.g., in the section introducing the Markov chain and the section deriving the eigenfunction) that the uniform innovations and boundary behavior at θ = −1 do not violate the required positivity or lead to absorption issues.
  2. [Generating functions and eigenfunction derivation] The explicit formulas for h and the persistence exponent are stated to follow from generating-function computations in the non-expanding regime. If these derivations rely on an unverified assumption that the principal eigenfunction is strictly positive on the entire domain, the resulting transition kernel may not be well-defined for all initial conditions; the paper should supply the precise domain on which h > 0 and confirm that the conditioned finite-dimensional distributions converge without additional boundary corrections.
minor comments (1)
  1. [Abstract] The abstract refers to 'phase-dependent structure in the parameters' without defining the phases; a brief sentence clarifying the phase boundaries within θ ∈ [-1,1) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and constructive suggestions. We address the two major comments below. Both points identify places where the manuscript would benefit from additional explicit verification; we will incorporate these clarifications in the revision.

read point-by-point responses

  1. Referee: [Markov representation and h-transform sections] The load-bearing step is the claim that the augmented chain (X_n, ε_n) satisfies the irreducibility, positivity, and regularity conditions needed for standard Doob h-transform theory on a continuous state space to apply directly and produce a positive eigenfunction h with simple principal eigenvalue. The abstract asserts that the representation 'admits an exact Markov representation' to which the theory applies, but the skeptic correctly notes that these conditions are not automatic for MA(1) representations; the manuscript must contain an explicit verification (e.g., in the section introducing the Markov chain and the section deriving the eigenfunction) that the uniform innovations and boundary behavior at θ = −1 do not violate the required positivity or lead to absorption issues.

    Authors: We agree that an explicit verification of the Harris irreducibility, positivity, and regularity conditions for the augmented chain (X_n, ε_n) is necessary for the direct application of Doob h-transform theory on a continuous state space. The current manuscript states that the representation is Markov and that the limiting distributions exist, but does not isolate a dedicated verification subsection. In the revision we will add a short subsection immediately after the definition of the augmented chain that checks: (i) the transition kernel is positive on a set of positive Lebesgue measure for any starting point in the interior, (ii) the uniform innovations ensure the chain is aperiodic and irreducible on the relevant half-space, and (iii) the boundary behavior at θ = −1 does not produce absorption (by exhibiting a uniform lower bound on the probability of returning to the interior in one step). These checks will be referenced again when the eigenfunction is constructed. revision: yes

  2. Referee: [Generating functions and eigenfunction derivation] The explicit formulas for h and the persistence exponent are stated to follow from generating-function computations in the non-expanding regime. If these derivations rely on an unverified assumption that the principal eigenfunction is strictly positive on the entire domain, the resulting transition kernel may not be well-defined for all initial conditions; the paper should supply the precise domain on which h > 0 and confirm that the conditioned finite-dimensional distributions converge without additional boundary corrections.

    Authors: The generating-function analysis yields an explicit candidate eigenfunction h that is strictly positive on the open set corresponding to the interior of the state space for θ ∈ [-1,1). We will add a precise statement of this domain (an open interval whose endpoints are determined by the phase of θ) together with a short argument that h remains positive and bounded away from zero on any compact subset of the interior. Because the persistence asymptotics already control the probability of hitting the boundary, the convergence of the conditioned finite-dimensional distributions follows from the standard h-transform construction without further boundary corrections; we will make this dependence explicit by citing the relevant persistence estimate in the convergence proof. revision: yes

Circularity Check

0 steps flagged

No circularity; standard Markov representation and explicit generating-function computations are self-contained.

full rationale

The paper states it represents the MA(1) model as a Markov chain on continuous state space, proves existence of limiting f.d.d. under positivity conditioning, and identifies the limit as a Doob h-transform. For θ ∈ [-1,1) it computes generating functions, extracts persistence asymptotics, and gives explicit formulas for eigenfunction h and the exponent. These steps are described as direct applications of standard theory plus explicit calculations on the given uniform-innovation model; no equations reduce a claimed prediction to a fitted parameter by construction, no self-citation chain is load-bearing, and no ansatz is smuggled via prior work. The derivation is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard results from Markov chain theory and Doob h-transforms; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of Markov chains and Doob h-transforms apply to the conditioned process
    Invoked when the model is represented as a Markov chain and the limit is identified as an h-transform.

pith-pipeline@v0.9.1-grok · 5653 in / 1255 out tokens · 60671 ms · 2026-06-28T13:00:57.569022+00:00 · methodology

0 comments
read the original abstract

We study an MA(1)-process with uniform innovations conditioned to stay positive. Representing the model as a Markov chain, we prove the existence of the limiting finite-dimensional distributions under this conditioning and identify the limiting process explicitly as a Doob $h$-transform. In the non-expanding case, i.e. when the coupling parameter $\theta$ satisfies $\theta\in[-1,1)$, we compute the relevant generating functions, extract sharp persistence asymptotics, and give explicit formulas for the eigenfunction $h$ and the persistence exponent. The resulting transition kernel of the limiting process is therefore fully explicit and displays a phase-dependent structure in the parameters. This provides a rare solvable example of a Markov chain on a continuous state space conditioned on persistence.

Figures

Figures reproduced from arXiv: 2606.01832 by Frank Aurzada, Virginia Worf.

Figure 1
Figure 1. Figure 1: Phase diagram of the main results Remark 3. The form of h in the trivial cases θx ≥ 1 and θx < −t follows directly from equation (3) and the continuity of h. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

34 extracted references · 4 canonical work pages · 2 internal anchors

  1. [1]

    Persistence for a class of order-one autoregressive processes and Mallows-Riordan polynomials.Adv

    Gerold Alsmeyer, Alin Bostan, Kilian Raschel, and Thomas Simon. Persistence for a class of order-one autoregressive processes and Mallows-Riordan polynomials.Adv. Appl. Math., 150:52, 2023. Id/No 102555

    2023

  2. [2]

    Persistence exponents via perturbation theory: Gaussian MA(1)-processes.Studia Mathematica, 283(3):257–280, 2025

    Frank Aurzada, Dieter Bothe, Pierre-Étienne Druet, Marvin Kettner, and Christophe Pro- feta. Persistence exponents via perturbation theory: Gaussian MA(1)-processes.Studia Mathematica, 283(3):257–280, 2025

    2025

  3. [3]

    Persistence exponents via perturbation theory: AR(1)- processes.J

    Frank Aurzada and Marvin Kettner. Persistence exponents via perturbation theory: AR(1)- processes.J. Stat. Phys., 177(4):651–665, 2019

    2019

  4. [4]

    Persistence exponents in Markov chains.Ann

    Frank Aurzada, Sumit Mukherjee, and Ofer Zeitouni. Persistence exponents in Markov chains.Ann. Inst. Henri Poincaré, Probab. Stat., 57(3):1411–1441, 2021

    2021

  5. [5]

    Persistence probabilities for MA(1) sequences with uniform innovations

    Frank Aurzada and Kilian Raschel. Persistence probabilities for MA(1) sequences with uniform innovations. Preprint, arXiv:2507.04427, 2025

    work page arXiv 2025

  6. [6]

    Persistence probabilities of MA(1) sequences with Laplace innovations andq-deformed zigzag numbers

    Frank Aurzada and Kilian Raschel. Persistence probabilities of MA(1) sequences with Laplace innovations andq-deformed zigzag numbers. Preprint, arXiv:2512.14152, 2025

    work page arXiv 2025

  7. [7]

    Persistence probabilities and exponents

    Frank Aurzada and Thomas Simon. Persistence probabilities and exponents. In: Andreas Kyprianou, René Schilling, and Thomas Simon, editors,Lévy Matters V, volume 2149 of Lecture Notes in Mathematics, pages 183-224. Springer, Cham, 2015

    2015

  8. [8]

    On conditioning a random walk to stay nonnegative.The Annals of Probability, 22(4): 2152–2167, 1994

    Jean Bertoin and Ron Doney. On conditioning a random walk to stay nonnegative.The Annals of Probability, 22(4): 2152–2167, 1994

    1994

  9. [9]

    Bray, Satya N

    Alan J. Bray, Satya N. Majumdar, and Grégory Schehr. Persistence and first-passage properties in nonequilibrium systems.Advances in Physics, 62(3): 225–361, 2013

    2013

  10. [10]

    Exponential convergence to quasi-stationary distribution and Q-process.Probab

    Nicolas Champagnat and Denis Villemonais. Exponential convergence to quasi-stationary distribution and Q-process.Probab. Theory Relat. Fields, 164(1–2):243–283, 2016

    2016

  11. [11]

    Probability and Its Applications

    Pierre Collet, Servet Martínez, and Jaime San Martín.Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems. Probability and Its Applications. Springer, Berlin, Heidelberg, 2013. 30

    2013

  12. [12]

    Klaus Deimling.Nonlinear Functional Analysis.Springer-Verlag, 1985

    1985

  13. [13]

    Ordered random walks and the Airy line ensemble.Probab

    Denis Denisov, Will FitzGerald, and Vitali Wachtel. Ordered random walks and the Airy line ensemble.Probab. Theory Relat. Fields, to appear, 2026

    2026

  14. [14]

    Persistence of autore- gressive sequences with logarithmic tails.Electron

    Denis Denisov, Günter Hinrichs, Martin Kolb, and Vitali Wachtel. Persistence of autore- gressive sequences with logarithmic tails.Electron. J. Probab., 27: Paper No. 154, 43, 2022

    2022

  15. [15]

    Denisov, D

    D. Denisov, D. Korshunov, and V . Wachtel. Markov chains onZ +: analysis of stationary measure via harmonic functions approach.Queueing Systems, 91(3–4):265–295, 2019

    2019

  16. [16]

    Denis Denisov, Dmitry Korshunov, and Vitali Wachtel.Markov Chains with Asymptoti- cally Zero Drift.Cambridge University Press, 2025

    2025

  17. [17]

    Conditional limit theorems for ordered random walks

    Denis Denisov and Vitali Wachtel. Conditional limit theorems for ordered random walks. Electron. J. Probab., 15: Paper No. 11, 292–322, 2010

    2010

  18. [18]

    Random walks in cones.Ann

    Denis Denisov and Vitali Wachtel. Random walks in cones.Ann. Probab., 43(3): 992– 1044, 2015

    2015

  19. [19]

    Persistence probabilities of autoregressive chains with continuous innovations

    Titouan Donnart and Thomas Simon. Persistence probabilities of autoregressive chains with continuous innovations. Preprint, arXiv:2604.05670, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  20. [20]

    Conditional Brownian motion and the boundary limits of harmonic func- tions.Bulletin de la Société Mathématique de France, 85: pp

    Joseph Doob. Conditional Brownian motion and the boundary limits of harmonic func- tions.Bulletin de la Société Mathématique de France, 85: pp. 431–458, 1957

    1957

  21. [21]

    Invariance principles for integrated random walks conditioned to stay positive.Ann

    Jetlir Duraj, Michael Bär, and Vitali Wachtel. Invariance principles for integrated random walks conditioned to stay positive.Ann. Appl. Probab., 33(1): 127–160, 2023

    2023

  22. [22]

    Invariance principles for random walks in cones.Stochas- tic Process

    Jetlir Duraj and Vitali Wachtel. Invariance principles for random walks in cones.Stochas- tic Process. Appl., 130 (7): 3920–3942, 2020

    2020

  23. [23]

    On certain class of entire functions and a conjecture by Alan Sokal

    Alexander Dyachenko. On certain class of entire functions and a conjecture by Alan Sokal. Preprint, arXiv:1309.7551, 2013

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  24. [24]

    Ordered random walks.Electron

    Peter Eichelsbacher and Wolfgang König. Ordered random walks.Electron. J. Probab., 13: 1307–1336, 2008

    2008

  25. [25]

    Persistence of one-dimensional AR(1)- sequences.J

    Günter Hinrichs, Martin Kolb, and Vitali Wachtel. Persistence of one-dimensional AR(1)- sequences.J. Theor. Probab., 33(1): 65–102, 2020

    2020

  26. [26]

    Krishna and Manjunath Krishnapur

    M. Krishna and Manjunath Krishnapur. Persistence probabilities in centered, stationary, Gaussian processes in discrete time.Indian J. Pure Appl. Math., 47(2): 183–194, 2016

    2016

  27. [27]

    A first passage time distribution for a discrete version of the Ornstein- Uhlenbeck process.J

    Hernán Larralde. A first passage time distribution for a discrete version of the Ornstein- Uhlenbeck process.J. Phys. A, 37(12): 3759–3767, 2004

    2004

  28. [28]

    Majumdar and Deepak Dhar

    Satya N. Majumdar and Deepak Dhar. Persistence in a stationary time series.Phys. Rev. E, 64:046123, 2001. 31

    2001

  29. [29]

    Morris, Alan Feldstein, and Ernie W

    Grainger R. Morris, Alan Feldstein, and Ernie W. Bowen. The Phragmén-Lindelöf prin- ciple and a class of functional differential equations. In: L. Weiss, editor,Ordinary Dif- ferential Equations: 1971 NRL–MRC Conference, pages 513–540. Academic Press, New York and London, 1972

    1971

  30. [30]

    Deutscher Verlag der Wissenschaften, 1963

    Nikola Obreskov.Verteilung und Berechnung der Nullstellen reeller Polynome. Deutscher Verlag der Wissenschaften, 1963

    1963

  31. [31]

    Cambridge Mathematical Library, 2nd edition, 2014

    Chris Rogers and David Williams.Diffusions, Markov Processes and Martingales, Volume 1, Foundations. Cambridge Mathematical Library, 2nd edition, 2014

    2014

  32. [32]

    Cambridge Mathematical Library, 2nd edition, 2000

    Chris Rogers and David Williams.Diffusions, Markov Processes and Martingales, Volume 2, Itô Calculus. Cambridge Mathematical Library, 2nd edition, 2000

    2000

  33. [33]

    On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states.J

    Eugene Seneta and David Vere-Jones. On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states.J. Appl. Probab., 3(2):403–434, 1966

    1966

  34. [34]

    Advanced Topics in Markov chains

    Jan Swart. Advanced Topics in Markov chains. Lecture Notes, 2023,https://staff. utia.cas.cz/swart/lecture_notes/chain23_11_06.pdf 32

    2023