Weighted L\'epingle inequality

Pavel Zorin-Kranich

Weighted L\'epingle inequality

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A stopping-time ladder proves weighted martingale variation bounds

desk verdict The paper's weighted Lépingle inequality is new and the proof is sound—the one flagged gap in Lemma 2.2 is a harmless wording issue, not a real gap. read the letter →

arxiv 1908.05937 v2 pith:GW47IASR submitted 2019-08-16 math.PR math.CA

Pavel Zorin-Kranich This is my paper · ORCID

classification math.PRmath.CA MSC 60G1760G42

keywords weightedvariationinequalityp-variationmartingaleA_pcharacteristicstoppingtimessquarefunctionBanachspacecotype

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

The paper establishes a weighted version of the classical pathwise $r$-variation estimate for martingales: for every $12$, the weighted $L^p(\Omega,w)$ norm of the $r$-variation $V_rX$ of a martingale is bounded by a constant times $\sqrt{r/(r-2)}\,Q_p(w)^{\max(1,1/(p-1))}\lVert X\rVert_{L^p(\Omega,w)}$, where $Q_p(w)$ is the martingale $A_p$ characteristic of the weight. The interest is in the method: the proof uses a stopping-time ladder that decomposes the path's jumps by scale and avoids real interpolation, which had been the standard tool for such estimates. Because the dependence on the weight characteristic is explicit and sharp near $r=2$, the result feeds directly into extrapolation arguments that convert scalar weighted estimates into vector-valued estimates. A pathwise corollary bounds $V_r^\rho$ by a dyadic sum of $\rho$-powers of sampled jumps for any $0<\rho

What carries the argument

The central object is a dyadic stopping-time ladder. For an adapted process $X$ with values in a metric space, define the past oscillation $M_t=\sup_{t''\le t'\le t} d(X_{t'},X_{t''})$ and, for each $m\ge2$, let $\tau^{(m)}_0=0$ and $\tau^{(m)}_{j+1}$ be the first time after $\tau^{(m)}_j$ at which the process has moved at least $2^{-m}M_t$ from its value at $\tau^{(m)}_j$. The key lemma shows that any variation increment whose size lies between $2\cdot 2^{-m}M_t$ and $4\cdot 2^{-m}M_t$ is dominated by one jump of this ladder, up to a factor $8$; summing over dyadic scales gives the pathwise bound that turns $r$-variation into a weighted sum of $\rho$-th powers of sampled jumps. Applied with $\rho=2$ to real-valued martingales, this writes $V_rX$ as an $\ell^2$ sum of square functions, and the weighted square-function estimate for each sampled martingale supplies the $A_p$ characteristic in the final constant.

What would settle it

Compute the stopping-time ladder for a simple discrete-time path, for instance two unit jumps separated by a small flat stretch, and check whether the key lemma's conclusion holds when the comparison times $t'<t$ are fixed integers that satisfy the ratio condition $2<d(X_{t'},X_t)/(2^{-m}M_t)\le4$. A single path where no ladder jump lies between $t'$ and $t$ while the inequality fails would show the lemma, as stated, is false and that the proof of the main theorem needs an additional argument.

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Extended reading notes

Core claim

On its own terms, the paper claims that the classical pathwise $r$-variation moment estimate for martingales survives insertion of a weight with a constant that depends only on $p$, $r$, and the martingale $A_p$ characteristic $Q_p(w)$. Concretely, for every $1<p<\infty$ there is a constant $C_p$ such that for all $r>2$, all filtered probability spaces, all weights, and all integrable functions $X$, the associated martingale satisfies $$\lVert V_rX\rVert_{L^p(\$\Omega$,w)}\le C_p\sqrt{\frac{r}{r-2}}\,Q_p(w)^{\max(1,1/(p-1))}\lVert X\rVert_{L^p(\$\Omega$,w)}.$$ The proof reduces this to a pathwise statement: for $0<\rho<r$, the $\rho$-th power of the $r$-variation is dominated by $8^\rho\sum_m 2^{-(m-2)(r-\rho)}\sum_j \lvert X_{\tau^{(m)}_{j-1}}-X_{\tau^{(m)}_j}\rvert^\rho$, where the $\tau^{(m)}_j$ are stopping times adapted to the path's oscillations. With $\rho=2$, each inner sum is a square function of a sampled martingale, for which sharp weighted estimates are already available; summing the geometric series in $m$ yields the factor $\sqrt{r/(r-2)}$. The same corollary recovers vector-valued $p$-variation estimates for martingales with cotype $\rho$, with dependence $r/(r-\rho)$.

Load-bearing premise

The load-bearing premise is that the two times compared in the key lemma are not stopping times—that is, not random times determined by the information available up to that moment—even though in the discrete-time setting of the main theorem every fixed time is a stopping time.

Editorial extensions

If this is right

Near $r=2$, the constant cannot be improved to any rate below $\sqrt{\log r/(r-2)}$, so the theorem's $\sqrt{1/(r-2)}$ dependence is optimal up to a logarithmic factor.
By martingale extrapolation, the scalar weighted estimate at one exponent $p$ automatically gives vector-valued $L^p$ estimates for all $1<p<\infty$, a standard corollary of having a bound with explicit weight dependence.
The pathwise corollary with $\rho=2$ plus the usual square-function estimate yields the unweighted $r$-variation inequality, and monotone convergence extends the weighted theorem to càdlàg martingales.
For martingales taking values in a Banach space with cotype $\rho$, the same stopping-time argument gives $\lVert V_rX\rVert_{L^p}\le C\,r/(r-\rho)\,\lVert X\rVert_{L^p}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension, not pursued in the paper, is to use the same ladder to count jumps at each scale and prove strong variational inequalities, since the decomposition controls each scale separately.
The paper's hints at Orlicz-space endpoints can be made precise: the geometric summation in $m$ suggests that any Young function growing near zero like $x^2/(\log x^{-1})^{1+\epsilon}$ should satisfy the endpoint estimate, a route that would also sharpen the lower-bound comparison.
The hidden stopping-time assumption could be tested by an approximation argument: perturb the fixed times $t,t'$ slightly so they are not stopping times, prove the key lemma for the perturbed times, and pass to the limit; if the limit fails, a genuinely discrete argument would be needed.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a clean, small paper that deserves a proper referee. The main theorem—weighted p-th moment estimate on r-variation of general martingales with constant C_p sqrt(r/(r-2)) Q_p(w)^{max(1,1/(p-1))}—is new and useful. The novelty is real: prior weighted results were dyadic or used real interpolation; the stopping-time argument here avoids that and works directly for arbitrary filtered spaces. Corollary 2.4 is a nice pathwise inequality, and the extension to Banach-space valued martingales in Remark 3.5 shows the technique has legs.

I read it because the Pith Report flagged Lemma 2.2. The proof says 'by ... the assumption that t,t' are not stopping times,' and that assumption is not in the statement and is false for deterministic times in discrete time. So you might think there is a gap. There isn't. Let τ_j be the largest stopping time ≤ t. Then for every s ∈ [τ_j, t], the hitting condition d(X_s, X_{τ_j}) ≥ 2^{-m} M_s is false, otherwise τ_{j+1} ≤ s ≤ t, contradicting maximality. So the strict inequalities in the displayed contradiction follow without any non-stopping-time assumption. The clause can simply be deleted. I verified the rest of Lemma 2.2 similarly: the M_{τ'} < M_t/2 case also works by a triangle inequality argument. So the main theorem is not threatened.

What is the paper's real value? It makes the dependence on r and on the A_p characteristic explicit, it simplifies Lépingle's classical proof even in the unweighted case, and it points to applications in rough paths and BDG-type estimates. The citation pattern is honest: the core external inputs—weighted square-function estimates from Thiele–Treil–Volberg/Domelevo–Petermichl and extrapolation from Domelevo–Petermichl–Wittwer—are clearly labeled, and the self-citations are confined to remarks. No fitting of parameters, no invented entities.

Weaknesses are minor. The Lemma 2.2 wording really should be fixed before publication; a referee will trip on it. The proof of Theorem 1.1 delegates p ≠ 2 to extrapolation, which is fine but means the paper is not fully self-contained on that point. The p=1 remark in Remark 1.5 is telegraphic, but it points to a specific external theorem.

My recommendation: send it out. A good referee will find the paper correct and can request the one-line clarification in Lemma 2.2. I would accept it after minor revision.

Referee Report

0 major / 4 minor

Summary. The paper proves a weighted version of Lépingle's inequality for the pathwise r-variation of a martingale. For every 1<p<∞ and r>2, the L^p(w) norm of V_r X is bounded by C_p sqrt(r/(r-2)) Q_p(w)^{max(1,1/(p-1))} times the L^p(w) norm of the terminal variable, where Q_p(w) is the martingale A_p characteristic. The proof constructs a family of stopping times adapted to dyadic oscillation thresholds, proves a pathwise bound for the r-variation in terms of square functions of the sampled martingale, and then applies known weighted square-function estimates together with Rubio de Francia extrapolation. The stated novelty is that the proof avoids real interpolation techniques.

Significance. If the result holds, it provides the first weighted Lépingle inequality for general martingales with the sharp growth rate of the constant as r→2, complementing the lower bound in Remark 1.3. The pathwise stopping-time estimate in Corollary 2.4 is a clean and potentially useful tool beyond the weighted setting, for instance for rough-path BDG inequalities and Banach-space-valued martingales. The proof is concise and mostly self-contained, resting on standard external theorems: weighted square-function estimates for differentially subordinate martingales and a known extrapolation theorem. The result is not circular and does not fit any free parameters. The main caveat is the small gap in the proof of Lemma 2.2 discussed below, which is readily repairable and does not affect the central derivation.

minor comments (4)

[Section 2, Lemma 2.2] In the proof of Lemma 2.2, the sentence 'By the hypothesis (2.2) and the assumption that t,t' are not stopping times' introduces an assumption that is absent from the lemma statement and is false in the discrete-time setting, where every fixed time is a stopping time. The assumption is unnecessary: since j is the largest index with τ_j ≤ t, for every s ∈ (τ', t] the hitting condition d(X_s, X_{τ'}) ≥ 2^{-m} M_s cannot hold, because then τ_{j+1} ≤ s ≤ t would contradict maximality. Applying this to s = t' and s = t yields the displayed chain without any non-stopping-time assumption. Please delete the erroneous phrase and insert the maximality argument.
[Section 2, Corollary 2.4] The sentence 'By the monotone convergence theorem, we may assume that X_n becomes independent of n for sufficiently large n' is misleading: 'independent' suggests probabilistic independence, while the intended meaning is that the sequence is eventually constant in n (or has finite variation up to the horizon). Please rephrase, for example as 'eventually constant in n'.
[Section 2, Lemma 2.3 and Corollary 2.4] The notation V_r^t(X_t(ω)) is used without explicit definition in Section 2, although the earlier definition (1.1) writes V_r X(ω). Please define V_r^t(X_t) as the r-variation over the appropriate time interval and state any finite-horizon assumptions used in Lemma 2.3 and Corollary 2.4.
[Section 3, Theorem 3.1] Theorem 3.1 is a central black box, stated with 'cf. [DP16]' and references to several papers. Please make the precise statement and source of the weighted square-function estimate unambiguous, including the exact dependence of the constant on Q_p(w), so that the reader can verify the application to the sampled martingales without consulting multiple preprints.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from an independent pathwise stopping-time inequality combined with external weighted square function estimates and extrapolation; the paper's self-citations appear only in remarks and do not carry the proof.

full rationale

The central result, Theorem 1.1, is proved by combining the pathwise inequality Corollary 2.4 with the weighted martingale square function estimate Theorem 3.1 and the extrapolation theorem from [DPW17]. Corollary 2.4 is derived in a self-contained way from the stopping times defined in (2.1), via Lemmas 2.2 and 2.3; no parameter is fitted and no conclusion of the paper is reused as an input. Theorem 3.1 is attributed to prior external work by Thiele-Treil-Volberg, Domelevo-Petermichl, and Lacey, and the square function estimates are independent of the present paper's conclusions. The self-citations in the paper, namely [Zor15] in Remark 1.3, [KZ19] in Remark 3.4, and [MSZ18] in Remark 3.5, are used only to discuss applications, comparisons, or growth-rate motivation, not to prove Theorem 1.1. The only fragile point identified by the reader, the phrase 'the assumption that t,t' are not stopping times' in the proof of Lemma 2.2, is a technical premise that is absent from the lemma statement, but it is not a circularity: the argument can be repaired by maximality of j, and the alleged circularity is not a reduction of the theorem to its own inputs. Accordingly, the derivation chain is self-contained against independent external benchmarks and the circularity score is 0.

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

The theorem introduces no fitted constants, no new entities, and no ad hoc assumptions beyond standard martingale and weight theory. The dependence on Q_p(w) is an input from prior weighted square function estimates, not a fitted quantity. The only noteworthy internal assumption is the unflagged non-stopping-time condition in Lemma 2.2.

assumptions (4)

domain assumption Weighted martingale square function estimate (Theorem 3.1): for every 1<p<∞, ||(Σ|X_j-X_{j-1}|^2)^{1/2}||_{L^p(w)} ≤ C_p Q_p(w)^{max(1,1/(p-1))}||X||_{L^p(w)}.
Invoked for each sampled martingale in the proof of Theorem 1.1; not proved here, cited to [TTV15], [Lac17], [DP19], [DP16].
domain assumption Rubio de Francia extrapolation theorem for martingales ([DPW17, Theorem 8.1]) reduces the claim to a single p.
Used in the proof of Theorem 1.1 to obtain all 1<p<∞ from a weighted L^p estimate.
standard math Khintchine's inequality and standard Banach space facts used to derive square function estimates from differential subordination.
Implicit in Theorem 3.1; standard.
standard math Monotone convergence and standard stopping-time properties for discrete-time martingales.
Used to justify truncation in Corollary 2.4 and sampling by stopping times.

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Cite this review

Pith. "Pith review of Weighted L\'epingle inequality." pith.science (2026). https://pith.science/paper/GW47IASR

@misc{pith2026190805937,
  author       = {Pith},
  title        = {Pith review of: Weighted L\'epingle inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GW47IASR}},
  note         = {Machine review of arXiv:1908.05937}
}

abstract

We prove an estimate for weighted $p$-th moments of the pathwise $r$-variation of a martingale in terms of the $A_{p}$ characteristic of the weight. The novelty of the proof is that we avoid real interpolation techniques.

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Works this paper leans on

2 extracted references · 2 canonical work pages

[306]

Sharp Weighted $L^2$ inequalities for square functions

mr: 544802 (cit. on p. 1). [BO18] R. Bañuelos and A. Os¸ ekowski. “WeightedL2 inequalities for square functions”. In: Trans. Amer. Math. Soc.370.4 (2018), pp. 2391–2422. arXiv:1603.07618. mr: 3748572 (cit. on p. 4). [Bou89] J. Bourgain. “Pointwise ergodic theorems for arithmetic sets”. In:Inst. Hautes Études Sci. Publ. Math.69(1989).Withanappendixbytheaut...

work page Pith review arXiv 2018
[1458]

Positive sparse domination of variational Carleson operators

arXiv: 1612.03028. mr: 3829751 (cit. on p. 1). [DL12] Y. Do and M. Lacey. “Weighted bounds for variational Walsh-Fourier series”. In:J. Fourier Anal. Appl.18.6 (2012), pp. 1318–1339.mr: 3000985 (cit. on p. 1). [DOP17] Y. Do, R. Oberlin, and E. A. Palsson. “Variation-norm and ﬂuctuation estimates for ergodic bilinear averages”. In:Indiana Univ. Math. J.66....

work page Pith review arXiv 2012

Pith tools

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

[1] [306]

Sharp Weighted $L^2$ inequalities for square functions

mr: 544802 (cit. on p. 1). [BO18] R. Bañuelos and A. Os¸ ekowski. “WeightedL2 inequalities for square functions”. In: Trans. Amer. Math. Soc.370.4 (2018), pp. 2391–2422. arXiv:1603.07618. mr: 3748572 (cit. on p. 4). [Bou89] J. Bourgain. “Pointwise ergodic theorems for arithmetic sets”. In:Inst. Hautes Études Sci. Publ. Math.69(1989).Withanappendixbytheaut...

work page Pith review arXiv 2018

[2] [1458]

Positive sparse domination of variational Carleson operators

arXiv: 1612.03028. mr: 3829751 (cit. on p. 1). [DL12] Y. Do and M. Lacey. “Weighted bounds for variational Walsh-Fourier series”. In:J. Fourier Anal. Appl.18.6 (2012), pp. 1318–1339.mr: 3000985 (cit. on p. 1). [DOP17] Y. Do, R. Oberlin, and E. A. Palsson. “Variation-norm and ﬂuctuation estimates for ergodic bilinear averages”. In:Indiana Univ. Math. J.66....

work page Pith review arXiv 2012