{"id":"3f4342c6-4fa5-44b4-8aad-9c616696630f","arxiv_id":"2607.20714","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"High minima of general Gaussian processes: scaled overshoots are exponential with mean the minimal covariance energy, and conditional minimizer laws converge to energy-optimal measures.","lead":"This paper proves that for any continuous Gaussian process on a compact space, conditioning on a high minimum makes the scaled gap to the threshold converge to an exponential distribution, with mean equal to the process's minimum covariance energy. It also shows that the location of the minimum converges to a measure that minimizes this energy, and gives explicit formulas for standard processes like fractional Brownian motion.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The paper proves clean, well-supported extensions of known results for smooth Gaussian processes to all continuous Gaussian processes on compact metric spaces. The main theorems are load-bearing and are demonstrated with a coherent RKHS/regression framework. The weakest assumption σ*^2>0 is clearly stated and its failure modes are honestly discussed. The proof of the tie lemma (13) is valid once the sign of the Cameron–Martin shift is handled carefully; the apparent gap resolves. The parallel remarks about unpublished companion [4] are not used in the proofs. No critical red flags are present, and the result is internally consistent; the correct verdict remains ACCEPT (unchanged).","tokens_in":14448,"tokens_out":34992,"duration_ms":252027,"concrete_test":"Re-run the proof of Lemma (13) with the explicit identity P(X∈B_0)=P(X-rh∈B_r) to confirm the Cameron–Martin transfer; as an independent computational check, simulate fBm H=0.3 on [1,2] with large u and compare the empirical conditional law of u(M-u) to Exp(σ_H^2) computed via the formula in Example 2.7.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims of Theorem 2.1 and Theorem 2.3 are supported by a coherent chain of reasoning. The main assumption σ*^2>0 is explicitly stated and is genuinely necessary: when σ*^2=0 the event {M>u} is empty (Lemma 2.2), and the paper gives examples where this occurs (fBm with 0 in the interval). This is a limitation, not a flaw. I examined the log-concavity proof, the regression/local-concentration arguments, and the Cameron–Martin differentiation in Theorem 2.3. The tie lemma (13) initially appears to have a gap in the Fubini/Cameron–Martin step, but the step is valid: if B_r={h nonconstant on argmin(X+rh)}, then P(X∈B_0)=P(X-rh∈B_r), and since X-rh has law equivalent to X, P_X(B_0)=P_X(B_r), so a.e.-r good shifts imply r=0 is good. Thus the proof of Theorem 2.3 is sound. I found no load-bearing mathematical error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the high-minimum regime of a centred continuous Gaussian process X on a compact metric space K. Let M = min_t X(t) and let σ_*^2 be the minimum covariance energy over probability measures on K. Assuming σ_*^2 > 0, the authors prove two main results. Theorem 2.1 shows that, conditionally on M > u, the scaled overshoot u(M-u) converges in distribution to an exponential random variable with mean σ_*^2 as u → ∞. Theorem 2.3 shows that every weak subsequential limit of the conditional law of a measurable minimizer T given M > u is an optimal covariance-energy measure; if the optimal measure is unique, the conditional law converges weakly to it. The proofs use standard Gaussian tools: Cameron–Martin shifts, log-concavity of Gaussian tails, RKHS regression, and Danskin’s formula. The paper also contains a self-contained Malliavin-type identity for the minimizer location (Section 3.5) and detailed examples for the Ornstein–Uhlenbeck process, fractional Brownian motion, and fractional Brownian sheet, including an explicit optimal measure for fBm in the appendix.","tokens_in":14704,"tokens_out":10180,"duration_ms":79749,"significance":"If the results hold, as the arguments indicate, this is a substantial extension of the smooth-process results of Chakrabarty and Samorodnitsky to general continuous Gaussian processes, with the right minimal hypothesis. The main theorems are proved from first principles: the RKHS optimality criterion is proven in Theorem 3.1, and the finite-dimensional asymptotics of Theorem A.1 are used only for motivation, not as a black box. The paper is refreshingly explicit about the necessary assumption σ_*^2 > 0 and the empty-conditioning case when σ_*^2 = 0 (Lemma 2.2 and the discussion after Theorem 2.1), including the fBm example with 0 in the interval. This is a genuine limitation but not a flaw. The explicit form of the optimal measure for fractional Brownian motion in Lemma A.2 is a valuable technical contribution. The proofs are detailed and the logical chain is coherent; I found no circularity or load-bearing error.","major_comments":[],"minor_comments":[{"comment":"The statement ‘Since M>u implies Y>u’ is used several times, but the one-line justification M ≥ Y - ||Z||_∞ (because m ≥ 1) is not written out. Adding it would make the regression step easier to follow.","section":"Section 3.1, after Eq. (9)"},{"comment":"The convexity upgrade from ψ(u)/u² → c to ψ(u+x/u)-ψ(u) → 2cx is correct, but the interval ordering needed for the monotonicity of secant slopes is implicit. Spelling out that u + x/u ≤ (1+ε)u for u ≥ sqrt(x/ε) would improve readability.","section":"Section 3.2, Eq. (12)"},{"comment":"The claim that the Ornstein–Uhlenbeck kernel is integrally strictly positive definite on [0,L], and hence that the optimal measure is unique, is stated without proof or reference. A short citation or a parenthetical justification would be helpful.","section":"Example 2.6"},{"comment":"The sentence ‘For each x this happens at most countably many r’ relies on concavity of φ_x in r; a brief parenthetical explaining that the non-differentiability set of a concave function is countable would make the argument self-contained.","section":"Section 3.3, tie lemma"}],"recommendation":"accept","confidential_remarks":"The paper is mathematically sound and well written. The companion manuscript [4] is cited for exact tail asymptotics, but the present results are proved independently, so there is no overlap concern. The appendix derivation of the optimal measure for fractional Brownian motion is careful and useful. No substantive revisions are needed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Emanuel, if you work on Gaussian extremes this is worth a serious look. The paper takes the known overshoot and minimizer-location limits for smooth Gaussian processes (Chakrabarty–Samorodnitsky) and proves the same conclusions for every continuous centred Gaussian process on a compact metric space, under the one genuinely necessary condition sigma_*^2>0. The overshoot u(M-u) conditional on M>u converges to exponential with mean sigma_*^2, and every weak subsequential limit of the conditional minimizer law is an optimal covariance-energy measure; uniqueness of that measure gives full convergence. The fBm example is the cleanest part: for H<1/2 they write an explicit density for the optimal measure on [a,b] with constant potential, and for H>=1/2 it's the point mass at a. The fBm sheet factorizes. Those explicit measures are a real deliverable.\n\nThe proofs are careful and mostly self-contained. The key moves—RKHS regression onto the optimal measure, log-concavity of the tail via Prékopa, and a Cameron–Martin tie lemma—are all handled with enough detail that I could check the steps. The tie lemma looks odd at first because it uses Fubini over r and then Cameron–Martin equivalence, but the argument is valid: a.e.-r good shifts push the bad set to a null set under the original law. The differentiation in Theorem 2.3 is justified by bounded difference quotients and atomlessness of M. I didn't find any load-bearing gap.\n\nSoft spots are modest. The condition sigma_*^2>0 is not automatic and, as the authors honestly say, when it fails the event {M>u} is empty for all u (e.g., fBm starting at 0). So the results don't apply to processes with a point of zero variance. That's a limitation, not an error. The paper also does not derive the tail asymptotics of P(M>u); that's the companion paper's job, and the finite-dimensional asymptotics from [3] are used only for motivation, so the self-citation is not a problem. The Malliavin section is optional but does give a selection-independent derivative D M = R_T, which is a nice aside.\n\nWho should read this: anyone working on exact asymptotics of Gaussian extremes, especially fBm and stationary processes. The paper deserves a serious referee; it advances a known result in a clean direction and ships explicit examples.","headline":"Extends smooth-process overshoot and minimizer-location limits to all continuous Gaussian processes on compact metric spaces, with honest treatment of the degenerate sigma*=0 case; the math looks right.","tokens_in":15128,"tokens_out":2684,"would_cite":true,"duration_ms":23738,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G15","60G70","60G17","60G22"],"pacs":[],"model":"deepseek-v4-flash","headline":"For continuous Gaussian processes, the scaled overshoot of a high minimum converges to an exponential law with mean equal to the minimum covariance energy, and the minimizer's conditional location converges to the optimal covariance-energy","keywords":["Gaussian process","infimum","argmin","weak convergence","reproducing kernel Hilbert space","covariance energy","overshoot","high minima"],"falsifier":"Simulate a stationary Ornstein–Uhlenbeck process with λ=1 and L=1 on [0,1]. The theorem predicts σ*²=2/3, an exponential overshoot with mean 2/3, and a limiting location mixture of point masses at 0 and 1 each with weight 1/3 plus a uniform density on (0,1) with weight 1/3. If, for thresholds u=2,4,6, the empirical conditional distribution of u(M-u) deviates systematically from Exp(2/3), or the location histogram does not approach that mixture, the central claim would be contradicted.","tokens_in":14370,"feed_emoji":"📉","tokens_out":6297,"duration_ms":49641,"temperature":0.7,"pith_summary":"This paper proves a universal limit theorem for the high minima of continuous Gaussian processes. For any such process on a compact metric space whose minimum covariance energy σ*² is strictly positive, it shows that, conditionally on the minimum M exceeding a high level u, the scaled overshoot u(M-u) converges in distribution to an exponential random variable with mean σ*². It further shows that every weak subsequential limit of the conditional law of the location of the minimum is an optimal covariance-energy measure, and if that measure is unique, the conditional law converges to it. These results extend earlier asymptotic findings from smooth Gaussian processes to all continuous Gaussian processes, with the exponential parameter identified as the solution of a quadratic energy minimization problem. The proof draws on the reproducing kernel Hilbert space of the process, log-concavity of the tail, and Cameron–Martin shifts.","feed_headline":"High Gaussian minima: overshoots become exponential","feed_subtitle":"The limiting mean equals the minimum covariance energy, and the argmin converges to the optimal energy measure.","key_machinery":"The central object is the covariance energy ER(μ) and its minimizer ν, with potential k_ν(t) = ∫ R(s,t) ν(ds). The argument uses the isonormal representation on the RKHS of R, the regression decomposition X = mY + Z with m = k_ν/σ*² and Y = ∫ X dν independent of Z, the log-concavity of q(u) (via Prékopa's theorem), a Cameron–Martin tie lemma showing every RKHS element is constant on the sample argmin set, and a Malliavin identity DM = R_T. These combine to reduce a high-level conditioning to a one-dimensional Gaussian computation, yielding the exponential overshoot and the location limit.","core_discovery":"The central claim is that the high-minimum regime of a continuous Gaussian process is governed entirely by the minimum covariance energy σ*² = min_μ ∫∫ R(s,t) dμ(s)dμ(t) and by the optimal measure(s) attaining it. Theorem 2.1 says L(u(M-u)|M>u) converges weakly to Exp(σ*²). Theorem 2.3 says any weak subsequential limit of L(T|M>u) is an optimal covariance-energy measure, so if the optimal measure is unique the conditional law converges to it. The proofs hinge on the regression decomposition X(t) = m(t)Y + Z(t) with Y = ∫ X dν and m = k_ν/σ*², which makes M a strictly increasing function of a single Gaussian, and on the log-concavity of the tail q(u) = P(M>u), which upgrades the quadratic tai","pith_inferences":["The exponential limit offers a practical rare-event approximation: compute σ*² by a quadratic energy minimization and then use the exponential law to estimate overshoot distributions, which is far cheaper than full path simulation.","The location result suggests that under a rare-event constraint the argmin of a Gaussian process concentrates on the support of the optimal measure, so that optimal measure acts as a 'prior' for the argmin in conditional inference problems.","The tie lemma—that every RKHS element is almost surely constant on the sample argmin set—is a standalone structural fact about Gaussian processes that may apply to other extremal questions, such as high-level excursions or suprema over subsets.","A natural next step, not addressed in the paper, is the rate of convergence in the exponential limit; the proof establishes the limit but leaves open whether the error decays like O(u^{-1}) or O(u^{-2}) depending on the process's regularity."],"forward_implications":["The exponential overshoot law holds for every continuous Gaussian process with σ*²>0, so no smoothness of sample paths is required.","The conditional minimizer location converges to the unique optimal covariance-energy measure when it exists; for the stationary OU process on [0,L] the limit is (δ₀+δ_L+λ dt)/(2+λL), and for fractional Brownian motion on [a,b] with H≥1/2 it is δ_a.","The parameter σ*², not the pointwise variance, controls both the tail rate and the overshoot scale, and the limiting location is asymptotically independent of the overshoot value.","The identity Cov(W(h), f(M)) = E[f'(M)h(T)] gives exact finite-u formulas, such as Cov(X(s),M) = E[R(s,T)].'","When σ*²=0 the event {M>u} has probability zero for all u, so the positivity assumption is sharp and cannot be removed."],"fun_headline_variants":["Minima overshoot: exponential limit with energy mean","Gaussian minima: overshoot exponential, minimizer energy-optimal","High minima: overshoot and argmin governed by energy","Overshoot of high minima: exponential with mean energy","Minimizer law converges to optimal energy measure"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The condition σ*²>0 is load-bearing: if the minimum covariance energy is zero, then P(M>u)=0 for every u, so the conditional laws are undefined—this happens, for example, for fractional Brownian motion when the index set contains 0.","fun_headline_variants_meta":{"raw":{"variants":["Minima overshoot: exponential limit with energy mean","Gaussian minima: overshoot exponential, minimizer energy-optimal","High minima: overshoot and argmin governed by energy","Overshoot of high minima: exponential with mean energy","Minimizer law converges to optimal energy measure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000722,"raw_usage":{"total_tokens":3081,"prompt_tokens":751,"completion_tokens":2330,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":2250}},"tokens_in":495,"tokens_out":2330,"duration_ms":14828,"temperature":1.0,"reasoning_tokens":2250,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T09:34:30.215533+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a stationary Ornstein–Uhlenbeck process with λ=1 and L=1 on [0,1]. The theorem predicts σ*²=2/3, an exponential overshoot with mean 2/3, and a limiting location mixture of point masses at 0 and 1 each with weight 1/3 plus a uniform density on (0,1) with weight 1/3. If, for thresholds u=2,4,6, the empirical conditional distribution of u(M-u) deviates systematically from Exp(2/3), or the location histogram does not approach that mixture, the central claim would be contradicted.","supporting_citations":[],"review_version":1}