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The H\'ajek-R\'enyi-Chow maximal inequality and a strong law of large numbers in Riesz spaces

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Pith's one-line read The paper proves a Riesz-space analogue of the Hájek–Rényi–Chow maximal inequality for submartingales and uses it to obtain a submartingale convergence theorem and Chow's strong law of large numbers.

desk verdict The main HRC maximal inequality in Riesz spaces is solid and new, but Theorem 6.3 (p>2 strong law) contains a Jensen-direction error at equation (6.7) and is not proved as printed; the flaw is localized and easily repaired. read the letter →

arxiv 1908.09012 v1 pith:Q2IZ3USX submitted 2019-08-23 math.FA math.PRmath.STstat.TH

Wen-Chi Kuo , David F. Rodda , Bruce A. Watson This is my paper · ORCID
classification math.FAmath.PRmath.STstat.TH MSC 46B4060F1560F25
keywords RieszspacesvectorlatticesHájek–Rényi–ChowmaximalinequalitysubmartingaleconvergencestronglawoflargenumbersconditionalexpectationoperatorsClarkson's
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

Classical probability's Hájek–Rényi–Chow inequality bounds a submartingale's running maximum by its weighted increments. This paper proves the same kind of maximal inequality in Riesz spaces—vector lattices with a conditional expectation operator, where martingales and order convergence take the place of measure-theoretic expectations and almost-sure limits. The inequality is the load-bearing tool for two applications: a submartingale convergence theorem, in which a non-negative $L^p$ submartingale satisfies $X_n/a_n\to 0$ in order whenever the weighted increments of $X_i^p$ are summable, and Chow's strong law of large numbers for martingale difference sequences in $L^p(T)$, $1

What carries the argument

The load-bearing object is the family of band projections $P_{(g-Y_i/a_i)^+}$. In a Dedekind complete Riesz space with weak order unit, each positive element $h$ generates a band, and $P_h f=\sup_n(f\wedge n h)$ is the projection onto that band; $P_{(g-Y_i/a_i)^+}$ projects onto the band where the weighted submartingale has exceeded the level $g$ at time $i$, and the product $U_n$ projects onto the band of the running maximum. Lemma 4.1 telescopes the differences of these projections into the positive increments $Y_{i+1}^+-Y_i^+$, converting a statement about running maxima into one about sums of increments. Two supporting identities carry the applications: the Riesz-space Jensen machinery, by which $X_i^p$ is again a submartingale when $X_i$ is a non-negative submartingale in $L^p(T)$, and the Riesz-space Clarkson inequality $|x+y|^p+|x-y|^p\le 2(|x|^p+|y|^p)$ for $1\le p\le 2$, which feeds the submartingale convergence theorem into the strong law.

What would settle it

Construct a Dedekind complete Riesz space with weak order unit, a strictly positive conditional expectation $T$, and a non-negative submartingale $(X_n,T_n)$ in $L^p(T)$ for which $\sum_{i=1}^\infty T_1[(X_{i+1}^p-X_i^p)/a_{i+1}^p]$ converges in order but $X_n/a_n$ does not converge in order; one such example would refute Theorem 5.1. A more targeted check is to test the cited power-submartingale lemma in a concrete space such as $L^p(\Omega,\mathcal{F},\mu)$ with conditional expectation $T$, since the convergence argument depends on that lemma.

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Extended reading notes

Core claim

The central claim is Theorem 4.2: if $(Y_i,T_i)$ is a submartingale in $L^1(T)$, $(a_i)$ is a non-decreasing sequence of positive real weights, and $g\in R(T_1)^+$, then $$T_1(I-U_n)g \le \frac{Y_1^+}{a_1}+\sum_{i=1}^{n-1} T_1\left[\frac{Y_{i+1}^+-Y_i^+}{a_{i+1}}\right],\quad U_n=\prod_{i=1}^n P_{(g-Y_i/a_i)^+}.$$ The left side measures the conditional expectation of the part of $g$ that remains above the running weighted maximum, while the right side is the weighted sum of positive increments of the submartingale. This is the Riesz-space form of the Hájek–Rényi–Chow maximal inequality; it specialises to Doob's maximal inequality and, through the power-submartingale route, yields an upcrossing-free proof that $X_n/a_n$ converges to zero in order under a weighted summability condition, plus Chow's strong law of large numbers for martingale difference sequences.

Load-bearing premise

The chain from the maximal inequality to convergence and the strong law rests on the cited Riesz-space Jensen result that the $p$-th power of a non-negative submartingale is again a submartingale; if that power property fails in some $L^p(T)$, the weighted convergence theorem and the strong law would not follow from the maximal inequality.

Editorial extensions

If this is right

  • Doob's maximal inequality in Riesz spaces follows as a special case of Theorem 4.2.
  • For a non-negative $L^p$ submartingale with $\sum_{i=1}^\infty T_1[(X_{i+1}^p-X_i^p)/a_{i+1}^p]$ order convergent, the weighted process satisfies $X_n/a_n\to 0$ in order, and the proof avoids upcrossing arguments.
  • For $1\le p\le 2$, a martingale difference sequence in $L^p(T)$ with $\sum_{i=1}^\infty T_1(|Y_i|^p/a_i^p)$ order convergent obeys $(1/a_n)\sum_{i=1}^n Y_i\to 0$ in order.
  • For $p>2$, the same strong law holds under the stronger moment condition $\sum_{i=1}^\infty T_1(|Y_i|^p/i^{1+p/2})<\infty$ with $a_n=n$, and under a Hölder-type condition for general weights.
  • A Riesz-space Clarkson inequality, $|x+y|^p+|x-y|^p\le 2(|x|^p+|y|^p)$ for $1\le p\le 2$, is established.

Reading between the lines

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

  • The same telescoping-of-band-projections argument is a natural template for convergence theorems in ordered vector spaces that lack an upcrossing theory; this is an extension the authors do not state.
  • The Riesz-space Clarkson inequality suggests that $L^p(T)$ for $1<p<2$ carries a lattice version of uniform convexity, which could open geometric questions about these spaces that the paper does not pursue.
  • One testable extension is to relax $g\in R(T_1)^+$ in Theorem 4.2, since that assumption enters through the commutation of $T_1$ with the band projection $P_g$; if commutation holds more broadly, the maximal inequality would cover more general levels.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper develops Riesz-space analogues of classical martingale inequalities. Theorem 4.2 establishes a Hajek-Renyi-Chow maximal inequality for submartingales in L1(T) using band projections onto the sets where the weighted process exceeds g. Theorem 5.1 gives an upcrossing-free weighted convergence theorem for non-negative submartingales in Lp(T). Section 6 applies these results to martingale difference sequences, obtaining Chow-type strong laws for 1 <= p <= 2 (Theorem 6.1 and Corollary 6.2) and for p > 2 (Theorem 6.3). Section 3 contains a Riesz-space Clarkson inequality for 1 <= p <= 2 and quotes a Riesz-space Burkholder inequality. The main technical novelty is the band-projection proof of the maximal inequality, which is then used as the engine for the convergence and strong-law theorems.

Significance. If the cited machinery from [10] is accepted, Theorem 4.2 is a genuine contribution: it is a clean, parameter-free transfer of the Hajek-Renyi-Chow maximal inequality to Riesz spaces, and it supports the submartingale convergence theorem and the strong laws that follow. The Clarkson inequality for Lp(T) in Section 3 is a useful independent result. The paper is written for specialists in Riesz-space stochastic processes and fits the journal's scope. The main limitation is that the proof of Theorem 6.3 as printed contains a genuine error in the Jensen step and a second unjustified bounding step. Both are localized and repairable, and the theorem itself appears to be true, but the p > 2 strong law is not proved until those repairs are made.

major comments (2)
  1. [§6, Theorem 6.3 proof, Eq. (6.7)] For p > 2 the map s -> s^{p/2} is convex, so Jensen's inequality in Riesz spaces ([10, Theorem 4.4]) gives T1[(sum_{i=1}^n |Y_i|^2)^{p/2}] >= (sum_{i=1}^n T1|Y_i|^2)^{p/2}, the reverse of the claimed (6.7). The displayed inequality is false in general: in scalar probability with p = 4 and independent Y1, Y2 having P(Y = ±1) = P(Y = ±2) = 1/4, E[(Y1^2+Y2^2)^2] = 29.5 > (E Y1^2 + E Y2^2)^2 = 25. Since (6.8) and (6.9) are derived from (6.7), the boundedness of Z_n, and hence the proof of Theorem 6.3, is not established as printed. The error is localized and repairable: the f-algebra inequality (sum_{i=1}^n |Y_i|^2)^{p/2} <= n^{p/2-1} sum_{i=1}^n |Y_i|^p yields (6.9) after applying T1, so the theorem's conclusion is recoverable.
  2. [§6, Theorem 6.3 proof after (6.9)] The step after (6.9) says that because T1|Xn|^p/n^p is order bounded by h, one has sum_{i=2}^{n-1}(1/i^p - 1/(i+1)^p)T1(|Xi|^p) <= p sum_{i=1}^{infty} i^{-p/2} h. This does not follow: the stated bound on T1|Xn|^p/n^p gives T1|Xi|^p <= i^p h, so the left side is at most p h sum_{i=2}^{n-1} 1/i, which is not bounded. A correct argument is available: with a_j = T1|Y_j|^p and R_i = i^{-(p/2+1)} sum_{j=1}^i a_j, (6.9) and the corrected (6.7) give T1|Xi|^p/i^{p+1} <= C R_i/i, and sum_i R_i/i converges because sum_i i^{-(p/2+2)} sum_{j=1}^i a_j = sum_j a_j sum_{i=j}^{infty} i^{-(p/2+2)} <= C sum_j a_j/j^{p/2+1}. Thus the proof needs revision at this step; it is another localized gap in Theorem 6.3.
minor comments (5)
  1. [§4, proof of Theorem 4.2] The sentence immediately after (4.8) states QiTi(Y_{i+1}^+ - Y_i^+) <= Ti(Y_{i+1}^+ - Y_i^+), but the displayed estimate needs T1Qi <= T1 applied to the positive differences; please rephrase, since the conclusion is correct but the written composition is confusing.
  2. [§6, proof of Theorem 6.1] After (6.5), the order-convergent sum should be sum T1(|X_{i+1}|^p - |X_i|^p)/a_{i+1}^p rather than the displayed sum T_i(|X_{i+1}|^p - |X_i|^p)/a_{i+1}^p, because Theorem 5.1 is stated with T1; the two versions agree after applying T1 = T1T_i.
  3. [§1] The paper defines strict positivity of T but the blanket assumption after the extension to L1(T) does not state that T (and hence T1) is strictly positive; Theorem 5.1 relies on strict positivity of T1, so add this hypothesis explicitly.
  4. [§3, Theorem 3.3] There is a typo in the statement: 'for all ∈ N' should be 'for all n ∈ N'.
  5. [§6] The notation Lp(T1) in Corollary 6.2 and Theorem 6.3 is not introduced; earlier definitions use Lp(T). Please make the initial conditional expectation T1 explicit or use a consistent notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main maximal inequality is derived within the paper, and the external results cited are independent.

full rationale

The paper's load-bearing chain is not circular. Theorem 4.2 is proved directly from Lemma 4.1, a telescoping inequality for band projections, together with the submartingale property Ti(Y_{i+1}^+) >= Y_i^+; neither the conclusion (4.5) nor any equivalent maximal inequality is fed into the proof. Theorem 5.1 uses [10, Corollary 4.5] only to know that powers of non-negative submartingales remain submartingales; that is an external result with stated assumptions not containing the convergence theorem, and the proof then applies the paper's own Theorem 4.2 and Kronecker's Lemma. The Remark after Theorem 5.1 candidly notes that an upcrossing-based proof could be obtained from self-cited [14, Theorem 3.5], but the actual proof does not rely on it. Sections 6 invoke Clarkson's inequality (proved in Section 3), Holder's inequality from [3]/[10], and Burkholder's inequality from [2]; these are independent published results, not renamed versions of the strong law. No fitted parameter is relabelled as a prediction, and no equation is by construction equal to its input. The potential flaw flagged at (6.7), where Jensen's inequality is invoked in a direction that is not valid for p>2, is a mathematical correctness issue rather than a circularity, and the surrounding derivation is not made circular by it. Hence no circular step is present.

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

Pure mathematics paper; no fitted parameters and no new entities. The proof rests on prior Riesz-space martingale machinery, including Jensen inequalities, Lp spaces, band projections, and one cited Burkholder inequality. The false Jensen direction in equation (6.7) is a proof error, not an invented entity.

assumptions (4)
  • standard math Standard band projection and principal band identities in Dedekind complete Riesz spaces, including P_g f = sup_n(f ∧ ng), commutation of products of band projections, and equations (4.1) through (4.3).
    Used throughout Section 4, in Lemma 4.1 and Theorem 4.2, citing [23] and proved in part in equations (4.1) through (4.3).
  • domain assumption Riesz-space Jensen inequality and the power-submartingale closure from [10, Corollary 4.5 and Theorem 4.4].
    Used in Theorems 3.2, 5.1, and 6.1 to pass from submartingales to powers and to bound conditional expectations.
  • domain assumption Strict positivity of the conditional expectation T1.
    Used in Theorem 5.1 to conclude from T1(f)=0 with f≥0 that f=0; this is assumed in the standing setup.
  • domain assumption Burkholder inequality in Riesz spaces from [2] with constants cp and Cp.
    Invoked as Theorem 3.3 and used in Theorem 6.3 to bound T1|X_n|^p via the quadratic variation.

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Pith. "Pith review of The H\'ajek-R\'enyi-Chow maximal inequality and a strong law of large numbers in Riesz spaces." pith.science (2026). https://pith.science/paper/Q2IZ3USX

@misc{pith2026190809012,
  author       = {Pith},
  title        = {Pith review of: The H\'ajek-R\'enyi-Chow maximal inequality and a strong law of large numbers in Riesz spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q2IZ3USX}},
  note         = {Machine review of arXiv:1908.09012}
}
abstract

In this paper we generalize the H\'ajek-R\'enyi-Chow maximal inequality for submartingales to $L^p$ type Riesz spaces with conditional expectation operators. As applications we obtain a submartingale convergence theorem and a strong law of large numbers in Riesz spaces. Along the way we develop a Riesz space variant of the Clarkson's inequality for $1\le p\le 2$.

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Works this paper leans on

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