{"id":"3b6cc864-ef8d-4469-8e13-bd1cb12937f3","arxiv_id":"2508.14202","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For inhomogeneous immigration processes, the paper proves convergence of the kth passage time and shows Yule (pure birth) immigration yields extreme value laws beyond the classical families.","lead":"This paper analyzes how long it takes for k of many searchers to hit a target when the searchers arrive over time, rather than all starting together. It proves limiting laws for two arrival models and finds that a pure birth arrival process produces extreme value distributions outside the classical Frechet, Gumbel, and Weibull families.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Abstract omits the tail/regular-variation hypotheses on the single-searcher passage-time distribution and the uniform-integrability step for moments; the central limit theorem is therefore unverified rather than refuted.","rationale":"The reader's weakest assumption already pointed at unspecified regularity conditions on the single-searcher passage-time distribution and at the coupling to the all-searchers-present process; I agree that this is the central gap. I add specificity: the missing condition is membership in the lower-tail domain of attraction (regular variation of the CDF near 0), and moment convergence additionally requires uniform integrability or an explicit L^p bound in the coupling. Because only the abstract is available, neither the presence of these hypotheses nor the correctness of the proof can be confirmed. The paper's claim is plausible and may be correct, so I do not REJECT it; I also do not see a demonstrated flaw strong enough to move the reader's UNVERDICTED verdict. Thus the correct verdict remains UNVERDICTED, i.e. UNCHANGED from the reader's assessment. The proposed test is the minimal check that would settle whether the concern lands: read the theorem hypotheses, or, if they are missing, exhibit the oscillating lower-tail counterexample.","tokens_in":663,"tokens_out":7545,"duration_ms":90638,"concrete_test":"Obtain the full text and inspect the main theorem (likely Theorem 2.1/3.1). First check that it states a normalization and a regular-variation/domain-of-attraction condition on the single-searcher passage-time distribution near its lower endpoint, e.g. P(T≤t) ~ t^α L(t) for α>0 and L slowly varying, and that the proof of moment convergence includes a uniform-integrability argument. If the condition is absent, run the following test: take a distribution whose lower tail is P(T≤t)=t(2+sin log(1/t)) for small t (extended arbitrarily away from 0), simulate Poisson immigration at rates 10^3, 10^4, 10^5, and plot the normalized kth passage time. If the histogram keeps changing with the rate, the missing regular-variation assumption is load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is a limit theorem for kth passage times as the immigration rate grows. Such a theorem has two indispensable ingredients: (i) an explicit normalization of the passage time (typically a time scale depending on the rate), and (ii) a domain-of-attraction/regular-variation condition on the single-searcher passage-time distribution F near its lower endpoint. The abstract supplies neither. Without (ii), the normalized minimum can fail to converge at all; for example, if F(x) behaves like x(2+sin log(1/x)) near 0, the quantity n F(t/n) oscillates and no nondegenerate limit exists. The separate claim of moment convergence is also nontrivial: distributional convergence alone does not imply convergence of moments, so the proof must show uniform integrability of the scaled passage times or an explicit moment bound. The announced coupling to the all-searchers-present process must therefore preserve not only the law of the minimum but also enough integrability to pass moments. This is a verification gap, not a demonstrated error; the full text may well contain exactly the missing hypotheses. But until they are checked, the strongest claim is undecided.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies kth passage times in inhomogeneous immigration processes, where searchers arrive over time either through a time-inhomogeneous rate or through a Yule pure-birth process. The abstract claims rigorous proofs of convergence in distribution and convergence of moments of the kth passage times as immigration rates grow, via a coupling to the all-searchers-present process, and a random time-shift representation for Yule immigration. It further claims that Yule immigration produces extreme value distributions outside the classical Fréchet-Gumbel-Weibull family, and draws comparisons to branching Brownian motion.","tokens_in":979,"tokens_out":1542,"duration_ms":18280,"significance":"If the results hold, they would extend extreme value theory to a class of dependent, non-identically distributed passage-time variables arising in biophysical search problems, and would provide one of few exact non-classical extreme value distributions for such variables. The coupling perspective is potentially useful. However, the abstract alone does not provide enough detail to assess correctness; the central limit theorems are asserted without stating normalization, regularity conditions, or proof structure.","major_comments":[{"comment":"The central convergence-in-distribution claim omits the normalization and the regularity/tail conditions on the single-searcher passage-time distribution F. A limit theorem for the minimum of an increasing number of searchers requires a domain-of-attraction condition near the lower endpoint; without it, the normalized minimum can fail to converge (e.g., if F(x) ~ x(2+sin log(1/x)) as x->0, the quantity n F(t/n) oscillates). The abstract needs to state these conditions explicitly; otherwise the claimed 'rigorous proof' cannot be verified.","section":"Abstract"},{"comment":"The claimed convergence of moments is nontrivial: distributional convergence alone does not imply moment convergence. The coupling to the all-searchers-present process must preserve not only the law of the minimum but also the integrability of the scaled passage times. The abstract gives no indication of how uniform integrability or moment bounds are established. This is a load-bearing gap in the announced results.","section":"Abstract"},{"comment":"The coupling and random-time-shift constructions are only described in words. In particular, the Yule process is said to be 'viewed as a time inhomogeneous immigration process with a random time shift,' but no domain restrictions or error estimates are given. A rigorous proof must show that the random time shift converges in the right sense after normalization and that the coupling error vanishes in the limit. Without this, the claimed equivalence is not established.","section":"Abstract"}],"minor_comments":[{"comment":"The phrase 'one of the few examples where extreme value distributions can be obtained exactly' may overstate the novelty unless a clear comparison to existing literature is provided.","section":"Abstract"},{"comment":"The comparison to branching Brownian motion results is mentioned but not summarized; a sentence stating the nature of the comparison (agreement, contrast, or extension) would help the reader.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"This review is based solely on the abstract, as the full text was not provided. The announced results may well be correct, but the abstract omits essential hypotheses (regular variation/domain of attraction, uniform integrability) and proof details. I recommend obtaining the full manuscript before a substantive decision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know the gist: this paper claims a rare exact answer for kth passage times when searchers arrive over time, with Yule immigration producing limit distributions outside the Frechet/Gumbel/Weibull families. That is genuinely new and worth a look. The coupling idea — relating time-inhomogeneous immigration to the all-searchers-present process, and viewing Yule as a random time shift — is plausible and elegant. The comparison with branching Brownian motion gives useful context. The authors have a track record in this area, and the abstract is written by people who know what theorems should look like.\n\nThe soft spot is exactly what the abstract does not tell you. A limit theorem for the normalized minimum requires a regular-variation/domain-of-attraction condition on the single-searcher passage-time distribution near zero. If the abstract omits it because the full paper states it properly, fine. But without it, the claim of convergence in distribution is unverified; there are well-known oscillating examples where n F(t/n) fails to converge. The moment convergence claim is also a separate step: distributional convergence alone does not imply convergence of moments, so the proof must show uniform integrability or an explicit bound. The stress-test note is right to flag these as verification gaps, not demonstrated errors. The full text may well contain everything needed; the abstract just can't support the rigor claim on its own.\n\nThe biggest limitation of my assessment is that I only have the abstract. I would not want to desk-reject this; the problem is meaningful, the approach is novel, and the conclusion about non-classical extreme families is the kind of result that could be important for applied probability. But I also would not bet on the theorems without seeing the hypotheses and the coupling argument in detail.\n\nWho is the paper for? People working on extreme values of dependent variables, biophysical search models, and branching processes. If the proofs are correct, it will be cited. My advice: send it to a serious referee, preferably someone who knows both the one-dimensional passage-time literature and the branching Brownian motion connections, and ask specifically whether the stated theorems hold under the stated assumptions. If the omissions are just presentational, it is an easy accept after minor revision.\n\nFor the reading group: I would wait until the full text is out, then take a look. I would not cite it before seeing the full proof.","headline":"An abstract that promises a genuinely new exact extreme-value result for non-iid passage times, but the proof details and regularity hypotheses are not visible, so the correctness verdict has to wait for the full text.","tokens_in":1342,"tokens_out":1027,"would_cite":false,"duration_ms":12672,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G70","60J80"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that kth fastest passage times converge for immigration and pure-birth arrivals, and that Yule processes generate new extremal laws.","keywords":["kth passage time","immigration process","Yule process","extreme value theory","coupling","random time shift","fast searchers","branching process"],"falsifier":"Simulate a fixed search domain with a known single-searcher passage time distribution, add searchers at a finite but increasing rate, and measure the kth passage time; if its distribution or empirical moments do not approach the paper's limiting law as the rate grows, the asymptotic claim fails. For the Yule case, compare the observed limiting histogram to the predicted non-classical distribution; close agreement with a Frechet/Weibull/Gumbel fit would contradict the departure claim.","tokens_in":649,"feed_emoji":"⏱️","tokens_out":3985,"duration_ms":41523,"temperature":0.7,"pith_summary":"The paper studies hitting times of a target by the kth fastest among many independent searchers, when searchers are not all present at time zero but arrive over time. For time-inhomogeneous immigration rates, it proves that as the arrival rate grows, the kth passage time converges in distribution and in moments, and that this limit is the same as if all searchers had been present initially. For Yule (pure-birth) immigration, it shows the same kind of convergence holds, but the limiting extreme-value distributions fall outside the classical Frechet/Gumbel/Weibull family. The proofs use a coupling that recovers the all-searchers-present process, and a random time-shift that turns a Yule process into a time-inhomogeneous immigration process. The result matters because it extends extreme-value theory to a non-independent, non-identically-distributed setting that arises naturally in biophysical search problems.","feed_headline":"Yule-born searchers break classical extreme value laws","feed_subtitle":"New limit theorems connect immigration schedules to all-searchers-present search, and Yule births give non-classical extremes.","key_machinery":"The two main tools are a coupling that embeds the immigrating process into a process with all searchers present from the start, and a random time-change that maps a Yule pure-birth process onto a time-inhomogeneous immigration process. These constructions let the authors transfer convergence results from the classical 'all present' setting to the immigration setting, and they yield explicit formulas for the limiting kth passage time distributions.","core_discovery":"The central claim is that for a fixed search process and target, if searchers arrive at a target's domain at a growing rate, the time of the kth arrival at the target becomes asymptotically independent of the detailed arrival schedule, determined instead by the single-searcher passage time distribution. When arrivals are governed by a time-inhomogeneous rate, the kth passage time converges in distribution to the same limit as the corresponding order statistic for a process in which all searchers are present at time zero. When arrivals follow a Yule birth process, the same convergence holds, but the limiting law differs: it is not a Frechet, Gumbel, or Weibull distribution. The paper also pro","pith_inferences":["If Yule immigration yields non-classical extremal laws, then fitting real search data with a classical extreme-value distribution would systematically mis-estimate tail probabilities; the shape of the arrival process (birth-driven versus time-dependent) could serve as a diagnostic for when classical fits break down.","The random time-shift representation suggests the method could extend to other point processes with a suitable time-change, such as clustered or bursty arrivals, as long as the all-searchers-present coupling remains valid.","The convergence results may generalize to other order-statistic functionals of the search process, and possibly to moving targets, provided the coupling can be constructed in those settings."],"forward_implications":["For biophysical search with time-inhomogeneous arrivals, the fastest-searcher passage time can be computed from the single-searcher distribution once arrival rates are large, without simulating the full arrival schedule.","Moment convergence implies that mean passage times and other averaged observables also converge, making the results directly usable in mean first-passage-time calculations.","Yule immigration gives a new family of extremal laws, so predictions based on the classical Frechet/Weibull/Gumbel trichotomy should not be assumed for birth-driven arrival processes.","The coupling provides a rigorous bridge from immigration models to the previously studied all-searchers-present models, so classical results apply after a suitable rescaling of the arrival rate."],"supporting_citations":[],"fun_headline_variants":["Yule birth searchers yield new extreme value laws","Searcher arrival schedule reshapes extreme value limits","Non-Frechet extremes from Yule searcher arrivals","Yule immigration breaks classical extreme value classes","Time-inhomogeneous searcher arrivals alter extreme value laws"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The results are asymptotic: immigration rates must grow without bound while the search process and target stay fixed, and the single-searcher passage time must satisfy regularity conditions that are not spelled out in the abstract.","fun_headline_variants_meta":{"raw":{"variants":["Yule birth searchers yield new extreme value laws","Searcher arrival schedule reshapes extreme value limits","Non-Frechet extremes from Yule searcher arrivals","Yule immigration breaks classical extreme value classes","Time-inhomogeneous searcher arrivals alter extreme value laws"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001801,"raw_usage":{"total_tokens":6931,"prompt_tokens":745,"completion_tokens":6186,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":6108}},"tokens_in":489,"tokens_out":6186,"duration_ms":44038,"temperature":1.0,"reasoning_tokens":6108,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T18:41:40.330470+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a fixed search domain with a known single-searcher passage time distribution, add searchers at a finite but increasing rate, and measure the kth passage time; if its distribution or empirical moments do not approach the paper's limiting law as the rate grows, the asymptotic claim fails. For the Yule case, compare the observed limiting histogram to the predicted non-classical distribution; close agreement with a Frechet/Weibull/Gumbel fit would contradict the departure claim.","supporting_citations":[],"review_version":1}