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arxiv: 2605.01093 · v1 · submitted 2026-05-01 · 🧮 math.PR

Some Results on the Central Limit Theorem for Subsequences in Banach Spaces

Deli Li , Han-Ying Liang This is my paper

Pith reviewed 2026-05-09 18:17 UTC · model grok-4.3

classification 🧮 math.PR
keywords central limit theoremBanach spacescotype 2subsequencesweak convergencepartial sumsi.i.d. random variablesprobability in Banach spaces
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The pith

In Banach spaces of cotype 2, the normalized partial sums converge weakly if and only if their normalized versions along any subsequence of indices do.

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

The paper proves an if-and-only-if statement for the classical central limit theorem in Banach spaces: when the space has cotype 2, weak convergence of S_n over square root n holds exactly when the same normalized sums converge weakly along every subsequence of indices m_n. This equivalence shows that, in such spaces, one cannot have convergence hiding along sparse indices while failing for the full sequence. The authors further show that if a subsequence converges weakly under the usual square-root normalization but the full sequence does not, then no other sequence of positive constants a_n can make the full sums S_n over a_n converge weakly to a non-degenerate limit. They conjecture that the equivalence breaks down as soon as the space lacks cotype 2. The result therefore ties the existence of a limiting Gaussian measure directly to a geometric property of the underlying space.

Core claim

Let {X, X_n; n >= 1} be i.i.d. B-valued random variables with partial sums S_n. Under the assumption that B is a Banach space of cotype 2, (S_n / sqrt(n)) converges weakly if and only if (S_{m_n} / sqrt(m_n)) converges weakly for any subsequence {m_n} of positive integers. In addition, if the subsequence version converges weakly while the full sequence does not, then (S_n / a_n) fails to converge weakly to any non-degenerate probability measure for every sequence {a_n} of positive reals.

What carries the argument

The cotype-2 property of the Banach space B, which forces the weak convergence of the normalized sums S_n/sqrt(n) to be equivalent to the same convergence along every subsequence of indices.

If this is right

  • Weak convergence of the normalized sums along the full sequence implies the same convergence along every subsequence.
  • Weak convergence along even one subsequence forces the same convergence for the full sequence.
  • If the full sequence fails to converge weakly under the standard normalization, no alternative scaling a_n can produce a non-degenerate weak limit once a subsequence is already known to converge under square-root normalization.
  • The limiting measure, when it exists, must be the same Gaussian measure for the full sequence and all its subsequences.

Where Pith is reading between the lines

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

  • In cotype-2 spaces it becomes sufficient to check the central limit theorem along a sparse subsequence such as m_n = 2^n to settle the question for the entire sequence.
  • The second theorem implies that any failure of the central limit theorem in these spaces is robust: it cannot be repaired by rescaling once a subsequence already satisfies the square-root normalization.
  • The conjecture suggests that spaces lacking cotype 2 may admit i.i.d. sequences whose normalized sums converge weakly along lacunary indices but diverge along the full sequence.

Load-bearing premise

The Banach space B has cotype 2.

What would settle it

A concrete Banach space without cotype 2, together with i.i.d. random vectors in it, for which S_n/sqrt(n) fails to converge weakly yet S_{m_n}/sqrt(m_n) converges weakly along some subsequence m_n.

read the original abstract

Let $\{X, X_{n}; n \geq 1 \}$ be a sequence of i.i.d. $\mathbf{B}$-valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. This note is devoted to study the classical central limitr theorem for subsequences of sums of i.i.d. $\mathbf{B}$-valued random variables. We show that, under the assumption that $\mathbf{B}$ is of cotype $2$ space, $\left(\frac{S_{n}}{\sqrt{n}} \right)_{n \geq 1}$ converges weakly if and only if $\left(\frac{S_{m_{n}}}{\sqrt{{m_{n}}}} \right)_{n \geq 1}$ converges weakly for a subsequence $\{m_{n}; ~n \geq 1\}$ of positive integers. We conjecture that this result is false if $\mathbf{B}$ is not of cotype $2$ space. In addition, we show that, if $\left(\frac{S_{m_{n}}}{\sqrt{{m_{n}}}} \right)_{n \geq 1}$ converges weakly for a subsequence $\{m_{n}; ~n \geq 1\}$ of positive integers. and $\left(\frac{S_{n}}{\sqrt{n}} \right)_{n \geq 1}$ does not converge weakly, then $\displaystyle \left(S_{n}/a_{n} \right)_{n \geq 1}$ does not converge weakly to a non-degenerate probability measure for any sequence $\{a_{n}; n \geq 1 \}$ of positive real numbers.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript claims that for i.i.d. B-valued random variables in a Banach space B of cotype 2, the normalized partial sums S_n / sqrt(n) converge weakly if and only if S_{m_n} / sqrt(m_n) converge weakly along some subsequence {m_n} of positive integers. It conjectures that the equivalence fails if B is not of cotype 2, and additionally proves that if the subsequence version converges weakly while the full sequence does not, then S_n / a_n fails to converge weakly to any non-degenerate limit for every sequence of positive normalizers {a_n}.

Significance. If the central equivalence holds, the result would give a structural characterization of the CLT in cotype-2 spaces, showing that cotype 2 prevents the existence of convergent subsequences of laws without convergence of the full sequence. The conjecture would establish sharpness of the assumption, and the additional result would rule out alternative normalizations in the non-convergence case. These are potentially useful extensions of classical CLT theory in Banach spaces, but their significance depends on a correct proof of the non-trivial direction.

major comments (3)
  1. [Abstract / Theorem statement] Abstract (and the main theorem statement): the phrasing 'converges weakly for a subsequence {m_n}' is ambiguous with respect to quantifiers. It is unclear whether the claimed equivalence is (i) for every subsequence or (ii) that existence of even one convergent subsequence forces convergence of the full sequence (S_n / sqrt(n)). The non-trivial direction is (ii), which must invoke the cotype-2 inequality to transfer tightness or characteristic-function convergence from the subsequence to all n; the manuscript must state the theorem with explicit quantifiers and isolate the step where cotype 2 is used.
  2. [Abstract] Abstract: the conjecture that the equivalence fails when B is not of cotype 2 is asserted without any supporting argument, counterexample sketch, or reference to a known construction in non-cotype-2 spaces. This leaves the sharpness claim unsupported.
  3. [Abstract / final claim] The additional result (if subsequence converges but full sequence does not, then no {a_n} works): the manuscript must show explicitly how this follows from the main equivalence or from the cotype-2 assumption, rather than treating it as an immediate corollary. Without the proof, it is impossible to verify whether the argument avoids circularity with the definition of weak convergence.
minor comments (2)
  1. [Abstract] Typos: 'central limitr theorem' should be 'central limit theorem'; 'cotype 2 space' should be 'cotype-2 Banach space' for consistency with standard terminology.
  2. [Abstract] Notation: the random variables are denoted {X, X_n; n ≥ 1}, but the second X is redundant; standard notation is {X_n; n ≥ 1} i.i.d. copies of X.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and valuable suggestions. We agree that clarifications are needed on quantifiers, the conjecture, and the derivation of the additional result. We will revise the manuscript accordingly and address each point below.

read point-by-point responses

  1. Referee: [Abstract / Theorem statement] Abstract (and the main theorem statement): the phrasing 'converges weakly for a subsequence {m_n}' is ambiguous with respect to quantifiers. It is unclear whether the claimed equivalence is (i) for every subsequence or (ii) that existence of even one convergent subsequence forces convergence of the full sequence (S_n / sqrt(n)). The non-trivial direction is (ii), which must invoke the cotype-2 inequality to transfer tightness or characteristic-function convergence from the subsequence to all n; the manuscript must state the theorem with explicit quantifiers and isolate the step where cotype 2 is used.

    Authors: We agree the abstract phrasing is ambiguous and will revise it to state explicitly: the sequence (S_n / sqrt(n)) converges weakly if and only if there exists a subsequence {m_n} such that (S_{m_n} / sqrt(m_n)) converges weakly. This makes clear that the non-trivial direction is existence of one convergent subsequence implying full convergence. In the body of the paper, we will add a sentence isolating the application of the cotype-2 inequality, which is used to pass from tightness (or convergence of finite-dimensional distributions) along the subsequence to the full sequence via the cotype-2 property controlling the moments or Rademacher averages. revision: yes

  2. Referee: [Abstract] Abstract: the conjecture that the equivalence fails when B is not of cotype 2 is asserted without any supporting argument, counterexample sketch, or reference to a known construction in non-cotype-2 spaces. This leaves the sharpness claim unsupported.

    Authors: We acknowledge that the conjecture is stated without a counterexample or reference in the current version. We will revise the abstract to note that the conjecture is motivated by known examples in the literature where, in Banach spaces without cotype 2, one can construct i.i.d. sequences for which the CLT holds along a subsequence but fails for the full sequence (e.g., constructions based on spaces with no cotype 2 such as those appearing in works on the necessity of cotype conditions for the CLT in Banach spaces). If a specific short sketch can be added without lengthening the note excessively, we will include it; otherwise we will flag it explicitly as a conjecture. revision: partial

  3. Referee: [Abstract / final claim] The additional result (if subsequence converges but full sequence does not, then no {a_n} works): the manuscript must show explicitly how this follows from the main equivalence or from the cotype-2 assumption, rather than treating it as an immediate corollary. Without the proof, it is impossible to verify whether the argument avoids circularity with the definition of weak convergence.

    Authors: We agree that the additional claim requires an explicit derivation rather than being presented as immediate. In the revision we will insert a short paragraph after the main theorem explaining the argument, which holds in general Banach spaces (independent of cotype 2): suppose S_{m_n}/sqrt(m_n) converges weakly to a non-degenerate limit while S_n/sqrt(n) does not. Assume for contradiction that S_n/a_n converges weakly to some non-degenerate mu for a sequence a_n. By the definition of weak convergence and properties of i.i.d. sums, one can extract a further subsequence along which the normalization must be asymptotically equivalent to sqrt(n) (up to constants), leading to a contradiction with the assumed non-convergence of the sqrt(n)-normalized sequence. This uses only the portmanteau theorem and the fact that different normalizers would produce incompatible limits or degeneracy, avoiding circularity. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained via cotype-2 property

full rationale

The paper states an if-and-only-if theorem: in a cotype-2 Banach space, weak convergence of S_n/sqrt(n) holds exactly when weak convergence of S_{m_n}/sqrt(m_n) holds along some subsequence m_n. One direction is immediate from the definition of convergence in the space of probability measures (any convergent sequence has convergent subsequences). The converse direction is non-trivial and is asserted to follow from the cotype-2 inequality controlling the sequence of laws; this is an external structural property of the Banach space rather than a redefinition or tautology. The abstract and description contain no fitted parameters renamed as predictions, no self-referential definitions, and no load-bearing self-citations whose content reduces to the target claim. The additional non-convergence result for other normalizers is likewise independent. The derivation chain therefore does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard i.i.d. assumption for B-valued random variables and the geometric property that B has cotype 2; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption The sequence consists of i.i.d. B-valued random variables
    Standard setup for the classical CLT in Banach spaces.
  • domain assumption B is a Banach space of cotype 2
    The key geometric assumption that enables the equivalence.

pith-pipeline@v0.9.0 · 5605 in / 1333 out tokens · 59461 ms · 2026-05-09T18:17:30.120137+00:00 · methodology

discussion (0)

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Reference graph

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