The maximum-entropy median-martingale

Rikhav Shah; Vilas Winstein

arxiv: 2605.03206 · v2 · submitted 2026-05-04 · 🧮 math.PR

The maximum-entropy median-martingale

Rikhav Shah , Vilas Winstein This is my paper

Pith reviewed 2026-05-11 02:21 UTC · model grok-4.3

classification 🧮 math.PR

keywords maximum-entropy walkmedian-martingalearcsine distributionBrownian motionstationary distributionrandom walkunit interval

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The pith

The maximum-entropy median-martingale walk on the unit interval has the arcsine distribution as its stationary distribution.

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

This paper examines the unique maximum-entropy random walk on the unit interval whose next state has median equal to the current state. It proves that this walk's stationary distribution is the arcsine distribution and supplies a proof that ties the result to the two classical arcsine laws for Brownian motion. The work also generalizes the martingale property to characterize a wider family of walks under analogous constraints. A sympathetic reader would care because the construction gives a simple discrete process whose long-run behavior mirrors continuous Brownian paths.

Core claim

The stationary distribution of the maximum-entropy median-martingale walk on the unit interval is the arcsine distribution. This is shown by a proof that makes explicit the link to the two classical arcsine laws satisfied by Brownian motion. The paper further generalizes the martingale notion and characterizes the stationary distributions of a larger class of walks that obey similar median-type conditions.

What carries the argument

The median-martingale property, requiring that the median of the next state equals the current state, used together with the maximum-entropy selection criterion to define the walk on the unit interval.

If this is right

The long-run distribution of the walk is exactly the arcsine distribution.
The construction directly illuminates the connection to the arcsine laws of Brownian motion.
Generalizing the martingale condition allows characterization of stationary distributions for a broader family of walks.
The proof technique relies on symmetry properties shared by the discrete walk and Brownian motion.

Where Pith is reading between the lines

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

The same symmetry might let discrete simulations approximate certain Brownian functionals without solving the corresponding PDEs.
The approach could be tested on walks defined on other bounded intervals or with different median-type constraints.
If the pattern holds, similar maximum-entropy constructions might recover other classical distributions tied to diffusion processes.

Load-bearing premise

A unique maximum-entropy walk on the unit interval exists and satisfies the median-martingale property.

What would settle it

Finding any walk on the unit interval that maximizes entropy subject to the median-next-state condition yet converges to a stationary distribution other than the arcsine distribution would disprove the claim.

read the original abstract

This short note explores the maximum-entropy walk on the unit interval that is a median-martingale. That is, the median of its next state is equal to its current state. The stationary distribution of this walk is the arcsine distribution, and we provide a proof that elucidates the connection to two classical arcsine laws for Brownian motion. The notion of a martingale is further generalized, and a larger class of walks is considered and similarly characterized.

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.

Desk Editor's Note private letter to a colleague

Short note gives a max-entropy median-martingale on [0,1] whose stationary measure is the arcsine law, with a proof tying it to the classical Brownian arcsine results.

read the letter

This note constructs the maximum-entropy Markov walk on the unit interval that obeys the median-martingale condition (median of the next state equals the current state) and shows its stationary distribution is the arcsine law. It supplies a proof that connects the construction to the two classical arcsine laws for Brownian motion and then generalizes the martingale idea to characterize a larger family of such walks. The main new piece is the explicit entropy-maximization setup under the median constraint and the resulting stationary-distribution claim. The proof is presented as elucidating the Brownian link without circularity, which is the part that actually adds value for a short note. The generalization keeps the same flavor but spreads the idea a bit wider. The paper does a reasonable job keeping the focus narrow and the argument self-contained. The weakest assumption is that a unique maximum-entropy walk exists and satisfies the median property; the stress-test sees no internal inconsistency or hidden gap in the stationarity argument, so that part looks okay on the surface. The broader class of walks receives less detailed treatment than the main example, but nothing there looks load-bearing. This is for probabilists who care about entropy methods, discrete martingales, or stationary measures tied to Brownian motion. A reader already working on arcsine laws or constrained random walks would pick up the construction and the proof connection quickly. It deserves peer review as a short note because the central claim is specific, the link to classical results is worth verifying in detail, and the work is formally grounded enough to merit referee time even if revisions are needed.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs a maximum-entropy Markov walk on the unit interval [0,1] obeying the median-martingale condition (the median of the next-state distribution equals the current state). It proves that the stationary distribution of this walk is the arcsine law and supplies a proof relating the construction to the two classical arcsine laws for Brownian motion. The paper further generalizes the martingale notion and characterizes a larger class of walks with analogous properties.

Significance. If the construction and proof hold, the work supplies a parameter-free characterization of the arcsine distribution as the stationary measure of a maximum-entropy median-martingale process, together with an explicit link to Brownian-motion arcsine laws. The generalization to a broader class of walks adds scope. These features would constitute a modest but clean contribution to the interface of entropy maximization, martingale theory, and diffusion processes.

minor comments (2)

[Abstract] The abstract states that a proof is provided but does not indicate the section or theorem number where the connection to the classical arcsine laws is established; adding a forward reference would improve navigability.
The generalization of the martingale concept is mentioned only briefly; a short dedicated paragraph or subsection outlining the precise extension would clarify the scope of the larger class of walks.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. There are no major comments to address.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a maximum-entropy Markov walk on [0,1] obeying the median-martingale condition and proves that its stationary measure is the arcsine law, with an explicit connection to the classical arcsine laws for Brownian motion. The derivation is presented as a characterization supported by a proof; no step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation. The central claim remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is based solely on the abstract; the central claim rests on the existence of the maximum-entropy median-martingale and standard properties of the arcsine distribution and Brownian motion.

axioms (1)

domain assumption There exists a unique maximum-entropy random walk on the unit interval satisfying the median-martingale property.
The paper explores and characterizes this walk, so its existence is presupposed.

pith-pipeline@v0.9.0 · 5353 in / 1332 out tokens · 54985 ms · 2026-05-11T02:21:49.517342+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

ρ(s) = 1/Z_p (2/(x^p + (1−x)^p))^{1/p} ... lim p→0 M_p = √(ab) ... recovers arcsine
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

median-martingale ... maximum-entropy walk ... ℓ_q-minimizing-martingales

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · 1 internal anchor

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Pacheco–Gonz \'a lez, C. G., Stoyanov, J., and Universio, N. A class of markov chains with beta ergodic distributions

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Pincheira, P. M. Shrinkage Based Tests of the Martingale Difference Hypothesis . SSRN Electronic Journal\/ (2006)

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Revuz, D., and Yor, M. Excursions. In Continuous Martingales and Brownian Motion . Springer Berlin Heidelberg, Berlin, Heidelberg, 1999, pp. 471--513

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Random motions, classes of ergodic markov chains and beta distributions

Stoyanov, J., and Pirinsky, C. Random motions, classes of ergodic markov chains and beta distributions. Statistics and Probability Letters 50 , 3 (Nov. 2000), 293–304

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Stabilizing the Splits through Minimax Decision Trees

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[1] [1]

On expected occupation time of Brownian bridge

Chi, Z., Pozdnyakov, V., and Yan, J. On expected occupation time of Brownian bridge . Statistics and Probability Letters 97\/ (Feb. 2015), 83–87

work page 2015

[2] [2]

Finite-sample distribution-free inference in linear median regressions under heteroscedasticity and non-linear dependence of unknown form

Coudin, E., and Dufour, J.-M. Finite-sample distribution-free inference in linear median regressions under heteroscedasticity and non-linear dependence of unknown form . The Econometrics Journal 12 , S1 (2009), S19--S49

work page 2009

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Iterated random functions

Diaconis, P., and Freedman, D. Iterated random functions. SIAM Review 41 , 1 (Jan. 1999), 45–76

work page 1999

[4] [4]

Sur certains processus stochastiques homog\'enes

L\'evy, P. Sur certains processus stochastiques homog\'enes. Compositio Mathematica\/ (1940), 283--339

work page 1940

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G., Stoyanov, J., and Universio, N

Pacheco–Gonz \'a lez, C. G., Stoyanov, J., and Universio, N. A class of markov chains with beta ergodic distributions

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Pincheira, P. M. Shrinkage Based Tests of the Martingale Difference Hypothesis . SSRN Electronic Journal\/ (2006)

work page 2006

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Excursions

Revuz, D., and Yor, M. Excursions. In Continuous Martingales and Brownian Motion . Springer Berlin Heidelberg, Berlin, Heidelberg, 1999, pp. 471--513

work page 1999

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Random motions, classes of ergodic markov chains and beta distributions

Stoyanov, J., and Pirinsky, C. Random motions, classes of ergodic markov chains and beta distributions. Statistics and Probability Letters 50 , 3 (Nov. 2000), 293–304

work page 2000

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Stabilizing the Splits through Minimax Decision Trees

Zhang, Z., and Luo, H. Stabilizing the Splits through Minimax Decision Trees . arXiv 2502.16758\/ (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026