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A Marcinkiewicz-Zygmund inequality and the Kadec Pe{\l}czyn\'ski theorem in Orlicz spaces

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Pith's one-line read This paper extends the Marcinkiewicz–Zygmund inequality to Orlicz and Lorentz spaces and proves a Kadec–Pełczyński-type criterion for Orlicz spaces: a determining sequence has an $\ell^2$-equivalent subsequence exactly when $\int…

desk verdict The new Kadec-Pelczynski claim in Orlicz spaces is plausible but undefined and underproved; the rest is a mostly correct, non-novel simplification of Astashkin. read the letter →

arxiv 2506.04025 v4 pith:WFZ4G7XT submitted 2025-06-04 math.FA math.PR

Istvan Berkes , Eduard Stefanescu , Robert Tichy This is my paper · ORCID
classification math.FAmath.PR MSC 26D1546B0946B2546E3060E15
keywords Marcinkiewicz–ZygmundinequalityKhinchinKadec–PełczyńskidecompositionOrliczspacesLorentzYoungfunctionnorminequalitiesrandomseriesinBanach
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

This paper aims to extend two classical tools of $L^p$ Banach space theory to Orlicz and Lorentz spaces. It proves Khinchin-type and Marcinkiewicz–Zygmund-type norm inequalities with constants depending only on the Young function or on the indices $(p,q)$. Its central result is a generalized Kadec–Pełczyński theorem: for a Young function $\psi$ with $x \le \psi(x) \le x^2$, a determining sequence of normalized, uniformly integrable random variables tending weakly to $0$ has a subsequence equivalent to the unit vector basis of $\ell^2$ precisely when $\int x^2\,d\mu(x)$ belongs to $L^{\sqrt{\psi}}$, where $\mu$ is a limit random measure. If correct, this gives a structural criterion for $\ell^2$-subspaces in these Orlicz spaces, extending the earlier $L^p$ result for $1 \le p < 2$.

What carries the argument

The machinery is the Orlicz (Luxemburg) norm defined by a Young function $\psi$, together with the embedding chain $L^2 \subset L^\psi \subset L^1$ that holds on finite measure spaces when $x \le \psi(x) \le x^2$. The transfer lemma from the $L^p$ paper [4] is inequality (6.1), which for i.i.d. sequences bounds the norm of Rademacher (or i.i.d.) sums between a constant times $(\sum a_i^2)^{1/2}$ and $(\sum a_i^2)^{1/2}$; the present proof asserts this same two-sided bound for Orlicz norms. The deciding object is the limit random measure $\mu$, and the membership test is whether $\int x^2\,d\mu(x)$ lies in $L^{\sqrt{\psi}}$.

What would settle it

Take $\psi(x)=x^p$ with $1<p<2$ and compare Theorem 3.5 with the known $L^p$ Kadec–Pełczyński classification from [4]; if the criterion $\int x^2\,d\mu(x) \in L^{p/2}$ ever disagrees with the $L^p$ classification for a determining sequence, the claimed transfer is false.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Kadec–Pełczyński dichotomy survives in Orlicz spaces whose Young function grows at least linearly and at most quadratically. Theorem 3.5 states that for a determining sequence $(X_n)$ with $\|X_n\|_{L_\psi}=1$, with $\{\psi(|X_n|)\}$ uniformly integrable and $X_n \to 0$ weakly, there exists a subsequence equivalent to the $\ell^2$ unit vector basis if and only if $\int_{\mathbb{R}} x^2\,d\mu(x)$ lies in $L^{\sqrt{\psi}}$, where $\mu$ is a limit random measure. The condition interpolates between the $L^1$ and $L^2$ endpoints: the second moment of the limit random measure must live in the smaller Orlicz space generated by $\sqrt{\psi}$. The proof carries the $L^p$ argument of the earlier paper [4] over to Orlicz norms by replacing the $L^p$ norm in inequality (6.1) with the Orlicz norm, using the embeddings $L^2 \subset L^\psi \subset L^1$.

Load-bearing premise

The load-bearing premise is that the earlier $L^p$ proof can be transplanted to Orlicz spaces by swapping the $L^p$ norm for an Orlicz norm in the key inequality, a transfer the paper asserts but does not demonstrate, and if this transfer fails the generalized Kadec–Pełczyński theorem collapses.

Editorial extensions

If this is right

  • In Orlicz spaces with $\psi$ between the identity and $e^{(\cdot)}$, Rademacher sums have $L^\psi$ norm comparable to their $L^2$ norm (Lemma 3.1).
  • In Orlicz spaces with $\psi(x) \le \psi(2x) \ll \psi(x)$, the $L^\psi$ norm of a sum of independent zero-mean variables is comparable to the $L^\psi$ norm of its quadratic variation $(\sum |X_n|^2)^{1/2}$ (Theorem 3.2).
  • The same Marcinkiewicz–Zygmund comparison holds in Lorentz spaces $L^{p,q}$ for $1<p<\infty$ and $1 \le q \le \infty$ (Theorem 3.4).
  • The generalized Kadec–Pełczyński theorem gives an exact $\ell^2$-subsequence criterion in terms of a limit random measure, covering the $L^p$ cases $1 \le p < 2$ as a special case (Theorem 3.5).
  • For $\psi(x)=x^p$ with $1 \le p < 2$, the Orlicz results collapse to the classical Khinchin and Marcinkiewicz–Zygmund inequalities and to the earlier $L^p$ Kadec–Pełczyński theorem.

Reading between the lines

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

  • An extension not attempted here would be a Lorentz-space version of Theorem 3.5; since the Lorentz inequalities in Section 3 are proved by the same $L^2/L^1$ norm comparison, the same transfer idea might apply on the scale $L^{p,q}$.
  • The condition $\int x^2\,d\mu(x) \in L^{\sqrt{\psi}}$ has clean endpoints: for $\psi(x)=x^p$ it asks for $L^{p/2}$ integrability of the second moment, while for $\psi$ approaching $x^2$ it approaches an $L^1$ condition; checking these endpoints against classical $L^p$ criteria would sharpen the admissible range of $\psi$.
  • Because the proof relies only on norm domination rather than special structure of Orlicz spaces, the theorem is plausibly valid in any Banach function space whose norm sits between $L^1$ and $L^2$, a wider class than Orlicz spaces.
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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

3 major / 5 minor

Summary. The paper states and proves Khinchin-type estimates for Rademacher sums in Orlicz and Lorentz spaces (Lemmas 3.1 and 3.3) and Marcinkiewicz–Zygmund inequalities for independent zero-mean variables in these spaces (Theorems 3.2 and 3.4). The main new result is Theorem 3.5, a Kadec–Pełczyński-type criterion in Orlicz spaces: under a normalization, uniform integrability, and weak convergence to zero, the existence of a subsequence equivalent to the unit vector basis of l2 is claimed to be equivalent to the membership condition (3.5). The proof of Theorem 3.5 is given only as a sketch that reduces to the authors' earlier Lp paper [4], and several objects appearing in its statement are not defined in the present paper.

Significance. If the first four results are correct, they provide short, direct proofs of useful inequalities for Rademacher and independent sums in Orlicz and Lorentz spaces, and the Lorentz-space formulation with explicit dependence on p and q is a nice feature. The potential significance of the paper, however, is concentrated in Theorem 3.5, which would extend a Kadec–Pełczyński-type dichotomy to a broad class of Orlicz spaces. As it stands, that advance is not verifiable: the statement uses undefined terms and an undefined function space, and the proof is a deferred sketch rather than a worked argument. The paper does not ship machine-checked proofs or reproducible code, but it does contain explicit constants and a coherent overall strategy for the Khinchin/Marcinkiewicz–Zygmund parts.

major comments (3)
  1. [Section 3, Theorem 3.5] The statement of Theorem 3.5 is not well-defined under the definitions supplied in Section 2. The terms 'determining sequence' and 'limit random measure' are never defined, and the symbol L^{√ψ} is never defined. Moreover, for an admissible Young function such as ψ(x)=x^p with 1<p<2, the function √ψ(x)=x^{p/2} is not convex, so L^{√ψ} is not an Orlicz space in the sense of Section 2 and the membership condition (3.5) has no clear meaning. The proof in Section 6 therefore cannot be checked as it stands.
  2. [Section 6, inequality (6.1)] The replacement of the L^p norm in (6.1) by an Orlicz norm is not justified. In (6.1) the lower constant is ||ξ||_{L^1} and the upper constant is ||ξ||_{L^2}; after replacing the middle norm by ||·||_{Lψ}, the hypotheses ||X_n||_{Lψ}=1 and uniform integrability of {ψ(|X_n|)} do not by themselves imply a uniform positive lower bound on ||ξ_{n_k}||_{L^1} or a uniform upper bound on ||ξ_{n_k}||_{L^2}. No argument in Section 6 derives such bounds from the limit-measure condition (3.5), so the asserted reduction to the already-proved inequality (6.1) is incomplete and the sufficiency direction of Theorem 3.5 is unsupported.
  3. [Section 6, proof of Theorem 3.5, necessity] The necessity part is deferred to [4] with the assertion that the relation ||(1/√N)Σ X_{m_k}||^p = O(1) 'remains valid under the Orlicz norm'. No proof of this transfer is supplied, the definition of the subsequence (m_k) is not given in this paper, and the role of the condition ∫x²dμ(x) ∈ L^{√ψ} in the necessity argument is not explained. Consequently this direction is also unverified.
minor comments (5)
  1. [Section 4, equation (4.8)] The displayed chain in (4.8) is sloppy: the term || |S_N|^{2/3} ||_{L6} equals ||S_N||_{L4}^{2/3}, not a quantity controlled directly by ||S_N||_{L2}; the intended argument is to use the classical Khinchin estimate ||S_N||_{L4} ≪ ||S_N||_{L2}. Please rewrite the line to make that step explicit.
  2. [Section 2, equation (2.3)] The sentence 'we have for ψ≪φ if and only if Lφ⊂Lψ' is incomplete: it should state that on a finite measure space, if ψ≪φ then Lφ embeds continuously into Lψ, with the norm inequality as written. The current phrasing does not define the direction or the role of φ.
  3. [Section 5, equation (5.2)] Several sums in (5.2) and nearby lines are indexed as 'i=n' or 'i=n' where they should be 'i=1'; please correct these summation indices throughout.
  4. [Abstract and Introduction] There are multiple typographical slips, including 'Marczinkiewicz' for 'Marcinkiewicz' and 'Orlicz sp aces' in the abstract. Please run a careful proofreading pass.
  5. [References] Reference [20] is a book review of Krasnosel'skii and Rutickii's monograph; it would be more appropriate to cite the original monograph directly for the Orlicz-space properties used in Section 2.

Circularity Check

2 steps flagged · score 4.0 of 10

The Khinchin and Marcinkiewicz-Zygmund parts are independent, but the central Kadec-Pelczynski theorem is not proved; its statement and proof are deferred to the authors' own earlier paper [4] via an asserted Lp-to-Orlicz transfer.

  1. self citation load bearing [Section 3, Theorem 3.5 statement]
    "In the following we prove an extension of the well-known Kadec-Pełczyński theorem, where we use the terminology of the first and last authors work [4]."

    The objects in Theorem 3.5 ('determining sequence', 'limit random measure', and the space L^{sqrt(psi)}) are never defined in this paper. The statement is therefore only interpretable through the authors' own prior paper [4]. The new theorem's formulation and even its meaningfulness are carried by a self-citation rather than by definitions given in the present work.

  2. self citation load bearing [Section 6, proof of Theorem 3.5]
    "Since by the above explanation the last relation remains valid under the Orlicz norm, the proof of the necessity part of Theorem 3.5 follows again the same way as in [4]."

    The proof of the new theorem consists of asserting that the L^p inequality (6.1) from [4] remains valid under an Orlicz norm because L^2 embeds into L^psi and L^psi embeds into L^1, and then declaring that both directions 'follow in the same way as in [4]'. No derivation is given. Inequality (6.1) involves ||xi||_1 and ||xi||_2, quantities not controlled by the hypothesis ||X_n||_{L^psi}=1, so the transfer is not a formal consequence of the paper's own equations. The central claim is supported only by the authors' self-cited paper [4].

full rationale

The Khinchin inequalities (3.1, 3.3) and the Marcinkiewicz-Zygmund inequalities (3.2, 3.4) are proved directly using Young's inequality, Hölder's inequality, symmetrization, and the classical Rademacher argument; no step in those proofs assumes the target result. The only candidate for circularity is Theorem 3.5, whose statement imports undefined terminology from the authors' prior paper [4] and whose proof explicitly defers to [4] ('follows again the same way as in [4]'). This is a load-bearing self-citation rather than a definitional equivalence: (6.1) is an L^p inequality with constants ||xi||_1 and ||xi||_2, while the Orlicz hypotheses only normalize ||X_n||_{L^psi}=1, so the asserted transfer is not demonstrated and may even fail for psi(x)=x^p with p<2. That is a rigor or completeness defect, not a circular derivation by construction, since the paper never assumes Theorem 3.5 itself. Accordingly, the circularity score is moderate: the central claim lacks an independent derivation, but the earlier sections are self-contained and the main issue is heavy self-reliance rather than an explicit equation reducing to itself.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; all implicit constants depend only on ψ, p, and q and are universal. The axioms are standard functional analysis facts plus a substantial reliance on the authors' prior paper [4] for the main theorem.

assumptions (5)
  • standard math Classical Khinchin inequality for Rademacher sums with all p-norms equivalent to the L2 norm.
    Used in the lower bound of Lemma 3.1 via ||S_N||_4 ≤ C ||S_N||_2 in (4.8) and in Lemma 3.3 for the upper bound; the classical result is cited as [10].
  • standard math Standard embedding and norm comparison properties of Orlicz and Lorentz spaces: (2.3), (2.4), (2.5).
    These properties are cited to [3] and [20] and are used throughout Sections 4 and 5.
  • standard math Berkes-Tichy Lp Kadec-Pelczynski theorem and the two-sided estimate (6.1) for i.i.d. sums in Lp with 1 ≤ p < 2.
    Theorem 3.5 is proved by asserting that the proof in [4] transfers to Orlicz spaces; the transferred inequality (6.1) is the key input.
  • standard math Symmetrization inequality for sums of independent random variables and equivalence of Orlicz norms of randomized versus original sums.
    Used in Step 3 of Theorem 3.2 and in Theorem 3.4; the first inequality in (5.2) is asserted to follow as in the original proof without details.
  • domain assumption Implicit Fubini-type property for Orlicz norms on product probability spaces.
    Step 2 of Theorem 3.2 applies the scalar Khinchin inequality (3.1) to random coefficients X_i and treats the product-space Lψ norm as comparable to an iterated norm; this property is not stated or proved.

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Pith. "Pith review of A Marcinkiewicz-Zygmund inequality and the Kadec Pe{\l}czyn\'ski theorem in Orlicz spaces." pith.science (2026). https://pith.science/paper/WFZ4G7XT

@misc{pith2026250604025,
  author       = {Pith},
  title        = {Pith review of: A Marcinkiewicz-Zygmund inequality and the Kadec Pe\lczyn\'ski theorem in Orlicz spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WFZ4G7XT}},
  note         = {Machine review of arXiv:2506.04025}
}
abstract

In this paper, we extend the Marcinkiewicz--Zygmund inequality to the setting of Orlicz and Lorentz spaces. Furthermore, we generalize a Kadec--Pe{\l}czy\'nski-type result -- originally established by the first and third authors for $L^p$ spaces with $1 \le p < 2$ -- to a broader class of Orlicz spaces defined via Young functions $\psi$ satisfying $x \le \psi(x) \le x^2$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Lacunary Series, Nonlinear Functionals and Banach Space Structure

    math.FA 2026-06 unverdicted novelty 5.0 of 10

    Extends asymptotic results for lacunary series to nonlinear functionals f_k and proves Kadec-Pelczynski type theorems in Orlicz spaces.

Reference graph

Works this paper leans on

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