A note on P\'olya urns: the winner may lead all the time

Svante Janson

A note on P\'olya urns: the winner may lead all the time

T0 review · 0 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In a two-colour Pólya urn, an initial leader with at least as large a replacement bonus stays ahead forever with positive probability.

desk verdict A clean, correct proof of a folklore Pólya urn fact; worth citing and worth a referee. read the letter →

arxiv 2506.14859 v1 pith:DTQ7XHOD submitted 2025-06-17 math.PR

Svante Janson This is my paper · ORCID

classification math.PR MSC 60C0560J8060F15

keywords Pólyaurnmodelbranchingprocesscontinuous-timeembeddinglimittheorempositiveprobabilitymajorityrandomproportions

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

Pólya urn processes are usually described by their large-time behaviour: one colour eventually wins almost surely when its replacement bonus is the largest. This note asks a sharper pathwise question: can the winner be ahead from the very first draw all the way to infinity? The central result is that, for two colours, if the urn starts with more black balls than white balls and the black replacement bonus is at least the white bonus, then the event that $B_n > W_n$ for every $n \ge 0$ has strictly positive probability. The proof embeds the discrete urn in continuous time, where the two colour counts are independent branching processes, uses the full support of their rescaled almost-sure limits to build a positive-probability tail, and then matches that tail to the finite prefix without ever losing the initial lead. In the equal-replacement case, the same argument covers the random-limit regime where the eventual dominant colour is random.

What carries the argument

The mechanism is the continuous-time branching-process embedding. Give every ball an independent $\operatorname{Exp}(1)$ clock, and when a ball of colour $i$ rings replace it by $m_i+1$ balls of the same colour, each with a fresh clock; observing this continuous-time process at successive ring times reproduces the urn exactly. In the embedding, the black and white populations are independent continuous-time branching processes, and the standard supercritical limit theorem gives $e^{-m_b t}B_t \to Y_b$ and $e^{-m_w t}W_t \to Y_w$ almost surely, where $Y_b$ and $Y_w$ are positive random variables with full support on $(0,\infty)$. That limit identity does the work: it implies $\mathbb{P}(Y_w/Y_b < 1) > 0$, so the ratio $W_n/B_n$ is eventually below 1 on a positive-probability set. The finite prefix is handled by observing that on a run of black draws followed by white draws, $B_n-W_n$ rises and then falls, so two positive endpoints force positivity throughout.

What would settle it

For a small admissible case, such as $m_b=m_w=1$, $b_0=2$, $w_0=1$, compute the exact probability that $B_n>W_n$ for all $n\ge0$; a value of $0$, or a proof that the hitting time $\inf\{n: B_n \le W_n\}$ is almost surely finite, would falsify Theorem 1. Since the theorem asserts that this probability is strictly positive, any exact calculation or exhaustive finite-state analysis showing it is zero for one admissible parameter set would settle the claim against the paper.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1: for a two-colour Pólya urn with replacement parameters $m_b \ge m_w$ and initial counts $b_0 > w_0 \ge 0$, there is a positive probability that $B_n > W_n$ for all $n \ge 0$. This is stronger than the classical almost-sure statement that the advantaged colour eventually wins, because it gives a positive-probability family of paths on which the advantaged colour is never behind. The proof shows that the eventual limit $Z = \lim_{n\to\infty} W_n/B_n$, which is $0$ when $m_b > m_w$ and $Y_w/Y_b$ when $m_b = m_w$, satisfies $\mathbb{P}(Z < 1) > 0$; hence with positive probability the ratio is eventually below 1, and conditioning on a reachable state at some large time $N$ completes the argument through the Markov property.

Load-bearing premise

The load-bearing premise is the unproved branching-process limit theorem: after rescaling, each colour's population converges almost surely to a positive random variable whose support is the whole positive half-axis, and it is this full support that makes the eventual ratio less than 1 with positive probability.

Editorial extensions

If this is right

For every admissible parameter pair, the eventual winner can be identified at time zero on a positive-probability family of paths; the advantaged colour is never overtaken, not merely eventually dominant.
When $m_b = m_w$, the limiting proportion of the two colours is random with support in $(0,1)$, and the theorem guarantees a positive-probability set of paths on which the initially larger colour's proportion exceeds $1/2$ at every finite time.
The finite-colour generalization in Remark 2 holds: if colour 1 starts with a majority or plurality and has the largest replacement parameter, there is positive probability that the majority or plurality is maintained at every time.
The proof's structure implies that the positive-probability event can be decomposed into a finite sign-preserving prefix and an infinite tail, so the conclusion is not a boundary artefact of a single draw.

Reading between the lines

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

The same tail-plus-prefix scheme should transfer to any urn scheme whose rescaled colour counts converge to non-atomic positive limit variables, so the qualitative conclusion is likely not confined to linear replacement rules.
In the equal-replacement case, applying the theorem at the first time the initially trailing colour overtakes gives a positive conditional probability that the overtaking colour leads forever from that time onward; the paper does not state this two-sided version.
A quantitative version, not attempted here, would lower-bound the probability of leading forever by estimating $\mathbb{P}(Y_w < Y_b)$; the proof only establishes that this probability is positive.
Monte Carlo simulation of a fixed instance, such as $m_b=m_w=1$, $b_0=2$, $w_0=1$, should show the empirical frequency of never losing the lead staying bounded away from zero as the horizon grows, matching the theorem's positivity claim.

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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, correct proof of a folklore result about Pólya urns, and it's worth taking seriously as a citable note. The theorem—if the initially leading color also has at least as large a replacement bonus, it keeps the lead at every time with positive probability—is exactly what the title says, and the proof is complete modulo one standard external theorem. The author says he doesn't know a reference, and I believe him; the result may have been observed before, but not stated in this form in the cited literature. That makes the novelty a clean formulation and proof rather than a new method, which is fine for a note.

The continuous-time embedding is the right tool. Equations (1) and (2) are the Athreya–Ney limit for supercritical Bellman–Harris processes: after rescaling by e^{-m_i t}, the populations converge a.s. to positive random variables with full support on (0,∞). Then P(Y_w/Y_b < 1) > 0 is immediate, and the step from a positive probability of limsup W_n/B_n < 1 to a fixed N with positive probability is just nesting of events. The finite-prefix construction is the clever bit: if you can reach a black-majority state at time N with positive probability, draw all black balls first then all white; the difference B_n − W_n rises then falls, but since both endpoints are positive it never hits zero. That part is explicit and checkable.

Soft spots are minor. The main load-bearing input is the full-support conclusion of the Athreya–Ney theorem; it's standard and true for these processes, but the note doesn't prove it. A referee might ask for a one-line justification or a citation with theorem numbers, which the note already gives. The edge case w0 = 0 makes the cited theorem's support statement literally false because the white branching process is identically zero, but the theorem is trivially true in that case; the proof should split it off. Not a real problem, just a small precision issue. The multi-color remark leaves details to the reader, which is acceptable for a remark.

Who this is for: anyone working with Pólya urns or models of persistent advantage, and teachers who want a short, clean example of a positive-probability event inside a limit theorem. It deserves a serious referee; I'd send it out. It won't change the field, but it's a solid, honest note.

Referee Report

0 major / 3 minor

Summary. The paper studies a two-colour Pólya urn with colour-dependent replacement numbers m_b and m_w. The main result, Theorem 1, states that if the urn starts with strictly more black than white balls and m_b ≥ m_w, then with positive probability the black count exceeds the white count at every future time. The proof embeds the urn in a continuous-time branching process, applies Athreya–Ney limit theorems to obtain a non-degenerate limit for the ratio W_n/B_n, and then constructs a finite-prefix bridge from the initial state to a state from which the future lead persists with positive probability. The paper also notes that the result and proof extend to finitely many colours.

Significance. The result is a clean and intuitively natural persistence property: an initial lead combined with a replacement advantage does not merely make the favoured colour almost surely win, it also gives a strictly positive probability of leading at all times. The argument is elementary after standard branching-process facts, is free of fitted parameters, and gives explicit structure through the finite-prefix bridge. The note is suitable for a probability journal as a short contribution; the proof is self-contained except for the cited Athreya–Ney limit theorems, which are standard and appropriate.

minor comments (3)

[Proof, equation (2)] As stated, (2) fails in the permitted case w_0 = 0, because then W_t ≡ 0 and the limiting random variable Y_w is 0 rather than positive. The theorem's conclusion is immediate in this case, so the proof should either treat w_0 = 0 separately at the start or explicitly restrict the limit statements to w_0 > 0.
[Proof, after equation (4)] The sentence 'P(Y_w < Y_b) > 0 and thus (4) implies P(Z < 1) > 0 for any m_b and m_w' conflates two regimes: when m_b > m_w, Z = 0 deterministically and no support assumption is needed, while the support assumption is only needed in the case m_b = m_w. Splitting these two cases would make the argument easier to follow.
[Proof, after equation (10)] The event that the first k_b draws are black and the next k_w draws are white is asserted to have positive probability; this is true, but only because reachability of (b_N, w_N) forces k_w = 0 when w_0 = 0, and otherwise guarantees at least one ball of the required colour at each draw. A one-sentence justification of this feasibility would improve rigour.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation uses standard external branching-process limit theorems and a finite-prefix path argument that does not assume the conclusion.

full rationale

The paper's derivation chain is self-contained relative to the standard Athreya–Ney branching-process limit theorems cited in equations (1)–(2). These theorems are external mathematical results (Athreya & Ney 1972), not prior claims by the present author, and they are used only to establish that the limit ratio Z has positive probability of being less than 1. The finite-prefix step (draw black for k_b draws, then white for k_w draws) constructs an explicit event of positive probability that keeps B_n > W_n for all n <= N, and the Markov property then concatenates this with the positive-probability conditional event from (10). No parameter is fitted, no prediction is renamed, and no conclusion is assumed as an input. The only delicate point is the case w0 = 0, for which the quoted support assertion 'Y_w > 0' is literally false; however the theorem's conclusion is then trivial because W_n = 0 for all n, so this is an edge-case omission in the proof wording, not a circular step. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The proof introduces no free parameters or invented entities. It relies on standard continuous-time branching process results; the only external input is the Athreya-Ney limit theorem, which is a standard mathematical result rather than an arbitrary postulate.

assumptions (3)

standard math The discrete Pólya urn process has the same distribution as the embedded continuous-time branching process observed at the times the exponential clocks ring.
Used in the first paragraph of the proof; follows from independence and memorylessness of Exp(1) clocks.
standard math For a continuous-time branching process with Malthusian parameter m_i, e^{-m_i t} Z_i(t) converges almost surely to a strictly positive random variable Y_i whose distribution has support on the whole positive half-axis.
Invoked via Athreya and Ney [1, Theorems III.7.1 and III.7.2] in equations (1) and (2); this is the load-bearing external result.
standard math From positive probability of limsup W_n/B_n < 1, there exists an integer N such that P(W_n/B_n < 1 for all n >= N) > 0.
Used to obtain equation (7); follows from the definition of limsup and nesting of the events as N increases.

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Cite this review

Pith. "Pith review of A note on P\'olya urns: the winner may lead all the time." pith.science (2026). https://pith.science/paper/DTQ7XHOD

@misc{pith2026250614859,
  author       = {Pith},
  title        = {Pith review of: A note on P\'olya urns: the winner may lead all the time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DTQ7XHOD}},
  note         = {Machine review of arXiv:2506.14859}
}

abstract

Consider a P\'olya urn where a drawn ball of colour $i$ is replaced together with a fixed number $m_i$ of balls of the same colour. We give a simple proof that if, for example, there are two colours and the urn starts with more balls of colour 1 than 2, and $m_1\ge m_2$, then there is a positive probability that there always will be more balls of colour 1 than colour 2.

Discussion (0). Continue with ORCID to comment.

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

First to reach $n$ game
math.PR 2025-06 conditional novelty 6.0 of 10

Exact Catalan-number formula for expected profit and sharply different limit laws for the first-to-n-wins game under constant, reinforced, and without-replacement urn regimes.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages · cited by 1 Pith paper

[1]

Athreya & Peter E

Krishna B. Athreya & Peter E. Ney,Branching Processes. Springer- Verlag, Berlin, 1972

work page 1972
[2]

Eggenberger & G

F. Eggenberger & G. P´ olya, ¨Uber die Statistik verketteter Vorg¨ ange. Zeitschrift Angew. Math. Mech.3(1923), 279–289

work page 1923
[3]

P´ olya, Sur quelques points de la th´ eorie des probabilit´ es.Ann

G. P´ olya, Sur quelques points de la th´ eorie des probabilit´ es.Ann. Inst. Poincar´ e1, 117–161, 1931. Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden Email address:svante.janson@math.uu.se URL:http://www.math.uu.se/svante-janson

work page 1931

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Athreya & Peter E

Krishna B. Athreya & Peter E. Ney,Branching Processes. Springer- Verlag, Berlin, 1972

work page 1972

[2] [2]

Eggenberger & G

F. Eggenberger & G. P´ olya, ¨Uber die Statistik verketteter Vorg¨ ange. Zeitschrift Angew. Math. Mech.3(1923), 279–289

work page 1923

[3] [3]

P´ olya, Sur quelques points de la th´ eorie des probabilit´ es.Ann

G. P´ olya, Sur quelques points de la th´ eorie des probabilit´ es.Ann. Inst. Poincar´ e1, 117–161, 1931. Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden Email address:svante.janson@math.uu.se URL:http://www.math.uu.se/svante-janson

work page 1931