What to Expect When You're Expecting

Mark Whitmeyer

arxiv: 2606.30400 · v1 · pith:TPZRAFDNnew · submitted 2026-06-29 · 🧮 math.PR

What to Expect When You're Expecting

Mark Whitmeyer This is my paper

Pith reviewed 2026-06-30 04:59 UTC · model grok-4.3

classification 🧮 math.PR

keywords marginal degreesum of random variablesexpectationjoint distributionsmarginal distributionsSimons theoremprobability theory

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The pith

The marginal degree of the sum of n random variables is ceiling of n over 2.

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

The paper defines the marginal degree of sums in dimension n as the smallest integer k such that the joint distributions of all subcollections of at most k coordinates determine the value of the expectation of their sum, whenever defined. It proves this degree equals ceiling of n over 2 for every n at least 2. The upper bound is taken from an existing theorem while the lower bound is obtained by explicit constructions of two laws that match on smaller marginals yet produce different sum expectations. A reader would care because the result identifies the least partial information sufficient to fix the sum expectation in a multivariate setting.

Core claim

For every n greater than or equal to 2 the marginal degree is ceiling of n over 2. The upper bound follows from a theorem of Simons from 1977. The lower bound is proved by constructing, for every 1 less than or equal to k less than ceiling of n over 2, two joint laws whose marginals of dimension at most k agree, but for which the corresponding expectations of the sum are defined and unequal.

What carries the argument

The marginal degree of sums, the smallest k such that joints of size at most k determine the sum expectation.

If this is right

For even n the degree equals exactly n/2.
For odd n the degree equals exactly (n+1)/2.
When n equals 4 the sum expectation is fixed by all two-dimensional marginals.
No smaller collection of marginals suffices in general.

Where Pith is reading between the lines

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

The same threshold may govern determination of the distribution of the sum rather than only its expectation.
The constructions could be adapted to test minimal data requirements for other linear functionals.
The result suggests a general pattern for how many coordinates are needed to control additive statistics.

Load-bearing premise

A theorem of Simons from 1977 applies directly to the marginal degree of sums as defined here.

What would settle it

Two explicit joint distributions on n variables that agree on every marginal of dimension at most floor((n-1)/2) yet have defined but unequal expectations for the sum.

read the original abstract

The marginal degree of sums in dimension \(n\) is the smallest integer \(k\) such that the joint distributions of all subcollections of at most \(k\) coordinates of a real-valued random vector \(\left(X_1,\ldots,X_n\right)\) determine the value of \(\E\left(X_1+\cdots+X_n\right)\), whenever this expectation is defined. For every \(n\ge2\), we prove that this marginal degree is \(\left\lceil n/2\right\rceil\). The upper bound follows from a theorem of Simons (1977). The lower bound is proved by constructing, for every \(1\le k<\left\lceil n/2\right\rceil\), two joint laws whose marginals of dimension at most \(k\) agree, but for which the corresponding expectations of \(X_1+\cdots+X_n\) are defined and unequal.

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

The paper pins the marginal degree of the sum exactly at ceil(n/2) for n >= 2, with an explicit construction proving the lower bound.

read the letter

The main point is that the smallest k where all k-dimensional marginals fix the expectation of the sum is ceil(n/2). The lower bound is new and comes from concrete joint distributions that agree on marginals up to any smaller k but have different sum expectations. The upper bound is taken from Simons 1977.

The construction is the part that works cleanly. It gives, for each n and each k below the threshold, two laws whose low-dimensional marginals match but whose sum expectations differ when they exist. That shows directly why fewer marginals are not enough and stands on its own without external results.

The softer spot is the upper bound. The abstract says it follows from Simons without further derivation, and the stress-test note correctly flags that the theorem's hypotheses may not line up exactly with the paper's focus on expectations of the sum. If the full text just cites the result without checking the conditions, a referee will need to verify the transfer. That is the only real point of friction.

This is a narrow result in mathematical probability for readers who care about how much joint information controls linear expectations. The construction is worth recording even if the scope stays small. It is not a broad advance, but the lower-bound evidence is direct and the claim is precise.

I would send it to peer review. The construction is self-contained and the citation can be checked in one step; a referee can settle the Simons applicability without much trouble.

Referee Report

1 major / 0 minor

Summary. The paper defines the marginal degree of sums in dimension n as the smallest k such that the joint distributions of all subcollections of at most k coordinates determine E[X1+⋯+Xn] whenever this expectation exists. It claims to prove that this marginal degree equals ⌈n/2⌉ for every n≥2. The upper bound is asserted to follow from Simons (1977); the lower bound is established by constructing, for each 1≤k<⌈n/2⌉, pairs of joint laws whose marginals of dimension ≤k agree but whose sum expectations (when defined) differ.

Significance. If the result holds, it gives a precise, dimension-dependent characterization of the minimal marginal information needed to fix the expectation of a sum. The lower-bound constructions are self-contained and explicit, supplying concrete counterexamples that directly support the sharpness claim. The upper bound, however, is not derived internally.

major comments (1)

[Upper bound argument] Upper bound argument (abstract and main proof): the manuscript states that the upper bound 'follows from a theorem of Simons (1977)' but provides no verification that the theorem's hypotheses match the paper's definition of marginal degree. The definition requires that agreement on all marginals of dimension ≤k implies equality of E[sum] precisely when the expectation is defined, and focuses on the sum functional. It is not immediate that Simons (1977) applies directly to this setting without additional argument or adaptation; this is load-bearing for the claimed upper bound of ⌈n/2⌉.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment point by point below.

read point-by-point responses

Referee: [Upper bound argument] Upper bound argument (abstract and main proof): the manuscript states that the upper bound 'follows from a theorem of Simons (1977)' but provides no verification that the theorem's hypotheses match the paper's definition of marginal degree. The definition requires that agreement on all marginals of dimension ≤k implies equality of E[sum] precisely when the expectation is defined, and focuses on the sum functional. It is not immediate that Simons (1977) applies directly to this setting without additional argument or adaptation; this is load-bearing for the claimed upper bound of ⌈n/2⌉.

Authors: We agree that the manuscript would benefit from an explicit verification that Simons (1977) applies to our definition of marginal degree. While the paper asserts that the upper bound follows from the cited theorem, we acknowledge that the connection is not spelled out in detail. Simons' result concerns conditions under which expectations of sums are fixed by lower-dimensional marginals when the expectation exists, which aligns with our setup. In the revised version we will add a short explanatory paragraph (or subsection) confirming the match between the theorem's hypotheses and our definition, thereby making the upper bound fully justified within the manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; upper bound from external 1977 theorem, lower bound via explicit independent constructions

full rationale

The paper proves the marginal degree equals ⌈n/2⌉ for n≥2. The lower bound is obtained by direct construction of pairs of joint laws that agree on all marginals of dimension ≤k < ⌈n/2⌉ yet yield unequal expectations for the sum (when defined). The upper bound is stated to follow from the external theorem of Simons (1977), whose author is distinct from the present paper. No equations reduce a claimed result to a fitted parameter or to a self-referential definition; no load-bearing self-citations appear; the cited result is external and not part of a self-citation chain. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition of marginal degree, the applicability of the external Simons theorem, and the existence of the constructed joint distributions with defined but unequal sum expectations. No free parameters or invented entities are introduced.

axioms (1)

standard math Standard axioms of probability theory for defining real-valued random vectors, joint distributions, and expectations when they exist
Invoked implicitly to make the definitions and statements meaningful.

pith-pipeline@v0.9.1-grok · 5661 in / 1091 out tokens · 42619 ms · 2026-06-30T04:59:44.533783+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references

[1]

John P. Nolan,Basic properties of univariate stable distributions, Univariate Stable Distributions: Models for Heavy Tailed Data, Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2020, pp. 1–23

2020
[2]

1, 157–158

Gordon Simons,An unexpected expectation, The Annals of Probability5(1977), no. 1, 157–158

1977

[1] [1]

John P. Nolan,Basic properties of univariate stable distributions, Univariate Stable Distributions: Models for Heavy Tailed Data, Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2020, pp. 1–23

2020

[2] [2]

1, 157–158

Gordon Simons,An unexpected expectation, The Annals of Probability5(1977), no. 1, 157–158

1977