High Minima of Gaussian Processes: Overshoots and Minimizer Locations

Enkelejd Hashorva; Svyatoslav Novikov

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

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2026-08-01 09:34 UTC pith:3Q244XTN

load-bearing objection 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.

arxiv 2607.20714 v1 pith:3Q244XTN submitted 2026-07-22 math.PR stat.APstat.ML

High Minima of Gaussian Processes: Overshoots and Minimizer Locations

Enkelejd Hashorva , Svyatoslav Novikov This is my paper

classification math.PR stat.APstat.ML MSC 60G1560G7060G1760G22

keywords Gaussian processinfimumargminweak convergencereproducing kernel Hilbert spacecovariance energyovershoothigh minima

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

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.

Core claim

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

What carries the argument

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.

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

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.

Where Pith is reading between the lines

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

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

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.

read the 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.

The 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.

Soft 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.

Who 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.

Referee Report

0 major / 4 minor

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.

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.

minor comments (4)

[Section 3.1, after Eq. (9)] 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 3.2, Eq. (12)] 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.
[Example 2.6] 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 3.3, tie lemma] 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.

Circularity Check

0 steps flagged

No significant circularity: main proofs are self-contained; self-citations are motivational only.

full rationale

The paper's central results, Theorem 2.1 and Theorem 2.3, are derived from first principles rather than from fitted parameters or self-referential assumptions. The overshoot limit is obtained from the log-concavity of q(u)=P(M>u) and the logarithmic tail estimate (8), which is proved in Section 3.1 using the RKHS regression decomposition X=m(t)Y+Z with Y independent of Z, the optimality criterion k_nu>=sigma*^2 (Theorem 3.1, proved in the paper), and the Cameron-Martin theorem (Lemma 2.2). No parameter is fitted to the conclusion; sigma*^2 is defined from the covariance kernel. The location theorem is proved by Cameron-Martin differentiation (identity (14)), the tie lemma (13) proved directly via Fubini and Cameron-Martin equivalence, the regression estimate (10), and the RKHS embedding argument leading to k_mu=k_nu. The optimality criterion in Theorem 3.1 is quoted from the literature but its short proof is included, so it is not an imported black box. The finite-dimensional asymptotics of [3] and the vector result of [2] are used only in the introduction for motivation and in the appendix as background; they are not used to prove Theorems 2.1 or 2.3. The companion paper [4] is mentioned but not relied upon. The assumption sigma*^2>0 is explicit, and the paper itself notes that sigma*^2=0 makes the conditioning event empty (Lemma 2.2 and the comment following Theorem 2.1), so this is a genuine limitation rather than a concealed input. No equation or fitted parameter is renamed as a prediction, and no load-bearing conclusion reduces to a self-citation. The analysis therefore finds no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 9 axioms · 0 invented entities

No free parameters are fitted: the examples' constants (A_H, D_H, etc.) are derived, not hand-tuned. The assumptions are the standard process setting plus σ*^2>0 and standard theorems from Gaussian analysis. No new entities are introduced.

axioms (9)

domain assumption σ*^2 > 0 (minimum covariance energy positive)
Central assumption in Theorems 2.1 and 2.3. Lemma 2.2 gives equivalent conditions; the paper notes P(M>u)=0 when σ*^2=0.
domain assumption X is a centred Gaussian process with continuous sample paths on a compact metric space K
Setting of the paper; path continuity is used for the C(K)-valued random element and Fernique integrability.
standard math Cameron–Martin theorem for Gaussian measures on C(K)
Used in Lemma 2.2 and in the Cameron–Martin tie lemma (13) to transfer statements between X and X+rh.
standard math Log-concavity of Gaussian measures (Prékopa–Leindler)
Used in the proof of Theorem 2.1 to show q(u)=P(M>u) is log-concave.
standard math Borell–TIS inequality
Used in Section 3.1 to bound the sup norm of the residual process Z.
standard math RKHS isometry W: H → Gaussian space
Central representation (5) used throughout the proofs.
standard math Measurable selection theorem (Kuratowski–Ryll-Nardzewski)
Provides the Borel minimizer selection T used in the statements and proofs.
standard math Danskin's theorem for min of affine perturbations
Used in the Cameron–Martin tie lemma (13) and in deriving (14).
standard math Malliavin calculus identities
Used only in optional Section 3.5; not needed for the main theorems.

pith-pipeline@v1.3.0-alltime-deepseek · 14196 in / 18520 out tokens · 151397 ms · 2026-08-01T09:34:30.215533+00:00 · methodology

0 comments

read the original abstract

Let $X(t)$, $t\in K$, be a centred Gaussian process with continuous sample paths on a compact metric space $K$, and let $M=\min_{t\in K}X(t)$. Let $\sigma_*^2$ denote the minimum covariance energy associated with $X$, and assume that $\sigma_*^2>0$. Motivated by the results of \cite{chakrabarty2018asymptotic} for smooth Gaussian processes, we show that, conditionally on $M>u$, the scaled overshoot $u(M-u)$ converges, as $u\to\infty$, to an exponential random variable with mean $\sigma_*^2$. Moreover, every weak subsequential limit of the conditional law of a measurable minimizer of $X$ is an optimal covariance-energy measure. In particular, if this measure is unique, then the conditional law converges weakly to it. The results are illustrated by stationary Gaussian processes, fractional Brownian motion, and fractional Brownian sheet.

discussion (0)

Reference graph

Works this paper leans on

16 extracted references

[1]

Asymptotic behaviour of high Gaussian minima,

A. Chakrabarty and G. Samorodnitsky, “Asymptotic behaviour of high Gaussian minima,”Stochastic Processes and their Applications, vol. 128, no. 7, pp. 2297–2324, 2018

2018
[2]

Extremes of vector-valued Gaussian processes,

K. D¸ ebicki, E. Hashorva, and L. Wang, “Extremes of vector-valued Gaussian processes,”Stochastic Pro- cesses and their Applications, vol. 130, no. 9, pp. 5802–5837, 2020

2020
[3]

Asymptotic properties of type I elliptical random vectors,

E. Hashorva, “Asymptotic properties of type I elliptical random vectors,”Extremes, vol. 10, pp. 175–206, 2007

2007
[4]

Exact asymptotics for high infima of Gaussian processes in finite-active regimes

K. D¸ ebicki, E. Hashorva, and S. Novikov, “Exact asymptotics for high infima of Gaussian processes in finite-active regimes.” Manuscript, 2026

2026
[5]

High minima of non-smooth Gaussian processes,

Z. Wu, A. Chakrabarty, and G. Samorodnitsky, “High minima of non-smooth Gaussian processes,”Elec- tronic Communications in Probability, vol. 24, pp. Paper No. 53, 1–12, 2019

2019
[6]

Percolation of strongly correlated Gaussian fields, I: Decay of subcritical connection probabilities,

S. Muirhead and F. Severo, “Percolation of strongly correlated Gaussian fields, I: Decay of subcritical connection probabilities,”Probability and Mathematical Physics, vol. 5, no. 2, pp. 357–412, 2024

2024
[7]

On the existence of paths between points in high level excursion sets of Gaussian random fields,

R. J. Adler, E. Moldavskaya, and G. Samorodnitsky, “On the existence of paths between points in high level excursion sets of Gaussian random fields,”The Annals of Probability, vol. 42, no. 3, pp. 1020–1053, 2014

2014
[8]

Large deviations for high minima of Gaussian processes with nonnegatively correlated increments,

Z. Selk, “Large deviations for high minima of Gaussian processes with nonnegatively correlated increments,” Statistics & Probability Letters, vol. 206, p. 110001, 2024

2024
[9]

Minimum-energy measures for singular kernels,

L. Pronzato and A. Zhigljavsky, “Minimum-energy measures for singular kernels,”Journal of Computa- tional and Applied Mathematics, vol. 382, p. 113089, 2021

2021
[10]

On logarithmic concave measures and functions,

A. Pr´ ekopa, “On logarithmic concave measures and functions,”Acta Scientiarum Mathematicarum (Szeged), vol. 34, pp. 335–343, 1973

1973
[11]

The theory of max-min, with applications,

J. M. Danskin, “The theory of max-min, with applications,”SIAM Journal on Applied Mathematics, vol. 14, no. 4, pp. 641–664, 1966

1966
[12]

Gaussian null sets and differentiability of lipschitz map on Banach spaces,

R. R. Phelps, “Gaussian null sets and differentiability of lipschitz map on Banach spaces,”Pacific Journal of Mathematics, vol. 77, no. 2, pp. 523–531, 1978

1978
[13]

Cube root asymptotics,

J. Kim and D. Pollard, “Cube root asymptotics,”The Annals of Statistics, vol. 18, no. 1, pp. 191–219, 1990

1990
[14]

Continuit´ e absolue de la loi du maximum d’un processus continu,

D. Nualart and J. Vives, “Continuit´ e absolue de la loi du maximum d’un processus continu,”C. R. Acad. Sci. Paris S´ er. I Math., vol. 307, no. 7, pp. 349–354, 1988

1988
[15]

Nualart,The Malliavin Calculus and Related Topics

D. Nualart,The Malliavin Calculus and Related Topics. Berlin: Springer, 2 ed., 2006

2006
[16]

Smoothness of the law of the supremum of the fractional Brownian motion,

N. Lanjri Zadi and D. Nualart, “Smoothness of the law of the supremum of the fractional Brownian motion,” Electron. Commun. Probab., vol. 8, pp. 102–111, 2003. Enkelejd Hashorva, Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland Email address:Enkelejd.Hashorva@unil.ch Svyatoslav Novikov, Department of Actuar...

2003

[1] [1]

Asymptotic behaviour of high Gaussian minima,

A. Chakrabarty and G. Samorodnitsky, “Asymptotic behaviour of high Gaussian minima,”Stochastic Processes and their Applications, vol. 128, no. 7, pp. 2297–2324, 2018

2018

[2] [2]

Extremes of vector-valued Gaussian processes,

K. D¸ ebicki, E. Hashorva, and L. Wang, “Extremes of vector-valued Gaussian processes,”Stochastic Pro- cesses and their Applications, vol. 130, no. 9, pp. 5802–5837, 2020

2020

[3] [3]

Asymptotic properties of type I elliptical random vectors,

E. Hashorva, “Asymptotic properties of type I elliptical random vectors,”Extremes, vol. 10, pp. 175–206, 2007

2007

[4] [4]

Exact asymptotics for high infima of Gaussian processes in finite-active regimes

K. D¸ ebicki, E. Hashorva, and S. Novikov, “Exact asymptotics for high infima of Gaussian processes in finite-active regimes.” Manuscript, 2026

2026

[5] [5]

High minima of non-smooth Gaussian processes,

Z. Wu, A. Chakrabarty, and G. Samorodnitsky, “High minima of non-smooth Gaussian processes,”Elec- tronic Communications in Probability, vol. 24, pp. Paper No. 53, 1–12, 2019

2019

[6] [6]

Percolation of strongly correlated Gaussian fields, I: Decay of subcritical connection probabilities,

S. Muirhead and F. Severo, “Percolation of strongly correlated Gaussian fields, I: Decay of subcritical connection probabilities,”Probability and Mathematical Physics, vol. 5, no. 2, pp. 357–412, 2024

2024

[7] [7]

On the existence of paths between points in high level excursion sets of Gaussian random fields,

R. J. Adler, E. Moldavskaya, and G. Samorodnitsky, “On the existence of paths between points in high level excursion sets of Gaussian random fields,”The Annals of Probability, vol. 42, no. 3, pp. 1020–1053, 2014

2014

[8] [8]

Large deviations for high minima of Gaussian processes with nonnegatively correlated increments,

Z. Selk, “Large deviations for high minima of Gaussian processes with nonnegatively correlated increments,” Statistics & Probability Letters, vol. 206, p. 110001, 2024

2024

[9] [9]

Minimum-energy measures for singular kernels,

L. Pronzato and A. Zhigljavsky, “Minimum-energy measures for singular kernels,”Journal of Computa- tional and Applied Mathematics, vol. 382, p. 113089, 2021

2021

[10] [10]

On logarithmic concave measures and functions,

A. Pr´ ekopa, “On logarithmic concave measures and functions,”Acta Scientiarum Mathematicarum (Szeged), vol. 34, pp. 335–343, 1973

1973

[11] [11]

The theory of max-min, with applications,

J. M. Danskin, “The theory of max-min, with applications,”SIAM Journal on Applied Mathematics, vol. 14, no. 4, pp. 641–664, 1966

1966

[12] [12]

Gaussian null sets and differentiability of lipschitz map on Banach spaces,

R. R. Phelps, “Gaussian null sets and differentiability of lipschitz map on Banach spaces,”Pacific Journal of Mathematics, vol. 77, no. 2, pp. 523–531, 1978

1978

[13] [13]

Cube root asymptotics,

J. Kim and D. Pollard, “Cube root asymptotics,”The Annals of Statistics, vol. 18, no. 1, pp. 191–219, 1990

1990

[14] [14]

Continuit´ e absolue de la loi du maximum d’un processus continu,

D. Nualart and J. Vives, “Continuit´ e absolue de la loi du maximum d’un processus continu,”C. R. Acad. Sci. Paris S´ er. I Math., vol. 307, no. 7, pp. 349–354, 1988

1988

[15] [15]

Nualart,The Malliavin Calculus and Related Topics

D. Nualart,The Malliavin Calculus and Related Topics. Berlin: Springer, 2 ed., 2006

2006

[16] [16]

Smoothness of the law of the supremum of the fractional Brownian motion,

N. Lanjri Zadi and D. Nualart, “Smoothness of the law of the supremum of the fractional Brownian motion,” Electron. Commun. Probab., vol. 8, pp. 102–111, 2003. Enkelejd Hashorva, Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland Email address:Enkelejd.Hashorva@unil.ch Svyatoslav Novikov, Department of Actuar...

2003