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Mauricio Herrera-Marín

Publications and source records attributed to Mauricio Herrera-Marín.

13 recordsLinked to original sources
arXiv2026

Where should the null live? Surrogate tests for clustering of threshold-derived air-quality indices, with application to PM2.5 exceedances in Chile

Air-quality and climate monitoring routinely convert native-resolution measurements into exceedance counts before testing whether high values cluster in time. A surrogate null can be imposed on the derived index or on the native series and then propagated through the same thresholding and aggregation. We show that the two placements can encode different scientific hypotheses and lead to opposite conclusions. In 31 Chilean PM2.5 monitoring stations (2018--2025), weekly clustering of days above the 50 ug m^-3 national 24-hour standard was significant at all stations under an index-level IAAFT null but at only two under a daily native-resolution null; a native generative null with seasonal marginals and short-memory Gaussian-copula dependence agreed with the native surrogate at 30 of 31 stations. Across thresholds spanning the WHO guideline (15 ug m^-3) to the Chilean episode-alert level (80 ug m^-3), index-only rejections increased from 8 of 31 to all eligible stations as monthly event probabilities became more seasonally concentrated, and equalizing monthly event probabilities at the 50 ug m^-3 level reduced index-only decisions from 29 to five. We formalize the issue as null transport through a lossy observation map: an exact aggregate-level equivalent of a native null exists if and only if the randomization kernel is lumpable with respect to that map. A within-fiber diagnostic, which perturbs native amplitudes while keeping the observed index exactly fixed, shows that this condition fails in the monitoring data: the native null distribution moved by a median of 18 times its Monte Carlo variability. Weekly clustering of PM2.5 exceedances in Chile is therefore fully compatible with seasonality plus short-memory daily persistence, and index-level surrogate rejections should not be read as evidence about daily concentration dynamics.

stat.AP↗
arXiv2026

Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics

Neural operators learn maps between function spaces, while hereditary network dynamics are described by Volterra resolvents with non-rational Laplace symbols. We introduce a fractional Laplace neural operator (fLNO) that embeds this structure in the learned map. For commuting excitation--Laplacian pairs, one block graph-spectral layer represents the full linear Volterra solution operator exactly. We establish an expressivity frontier for finite rational realizations: they approximate fractional memory geometrically on compact frequency windows, but cannot reproduce the non-integer critical asymptotics generated by a branch point, and on the half-line the best rational rate is root-exponential. The same theory yields trainable parametrizations that enforce a prescribed stability margin by construction, and a graphon-transfer theorem separates genuine operator consistency from parameter sharing. In a common-data benchmark, positive rational operators can match or exceed fLNO accuracy on finite horizons, whereas in controlled near-critical experiments fLNO recovers the branching coordinate more faithfully with far fewer parameters; unconstrained rational fits can cross the stability boundary, while certified parametrizations cannot. A four-parameter spectral law transfers without retraining from graphs of size 48 to 192 with 0.51--0.62% relative error. Applications to Chilean aftershock sequences and to renewal models for Chile and 21 Italian regions illustrate structured inference with explicit uncertainty. The contribution is an operator-learning architecture in which exact memory structure, physical coordinates and stability guarantees coexist with competitive accuracy.

stat.ML↗
arXiv2026

Minimax and adaptive inference for branching ratios under long memory

The branching ratio $ρ$ of a self-exciting point process determines the branching margin $1-ρ$ to the critical boundary. We study inference on $ρ$ from a finite record for stationary Hawkes processes with completely monotone kernels whose slow relaxation mass obeys an envelope near zero rate. Slow excitation can then be exchanged for exogenous immigration; a spectral bound on the Kullback--Leibler rate and a coupling argument for the unobserved past give a minimax lower bound of order $T^{-γ/(2γ+1)}$ for a power-law envelope with tail exponent $γ$. A block-dispersion estimator whose block length grows as $T^{1/(2γ+1)}$ attains this rate. A quantitative central limit theorem, obtained from the cluster representation and a second-order Poincaré inequality on Poisson space, yields honest confidence intervals of nearly minimax expected length. For unknown $γ$, a Lepski-type rule adapts in probability at a logarithmic cost; free adaptation between memory classes is impossible, and honest intervals cannot adapt to lighter memory. Simulations are consistent with the rates.

math.ST↗
arXiv2026

Learning the exogenous rate but not the distance to criticality in nearly unstable heavy-tailed Hawkes processes

In the nearly unstable heavy-tailed regime of Jaisson and Rosenbaum, where the intensity of a linear Hawkes process converges to a rough square-root Volterra process, we ask what a record of length $T$ reveals about the relative distance to criticality $1-ρ$ and the exogenous rate $μ$. For the first, the information is governed by $\mathcal I_T=Tμ(1-ρ)$, not by the number of events; $\mathcal I_T$ stays bounded although the event count diverges like $T^{2γ}$, so no estimator of the relative distance to criticality is uniformly locally consistent. For the second, the Fisher information diverges because the rough limit has an atom at zero: the asymptotic information geometry is singular in the exogenous direction and, unlike the light-tailed case, has no Feller-type threshold. We prove that the joint maximum likelihood estimator of $(μ,ρ)$, whose likelihood is jointly concave, recovers $μ$ consistently at the random rate $\{Q_T/\log\log Q_T\}^{1/2}$ given by the observed information $Q_T$, while its estimate of the relative margin is only tight: the exogenous rate is learned although the endogenous parameter it is coupled to is not. For empty-start records $Q_T$ diverges because the rough limit started at zero spends positive time at zero; on windows observed long after the start, still with the branching ratio unknown, learnability holds with probability tending to one as the window grows, through the ergodicity of the rough limit and the atom of its stationary law. We also prove finite-dimensional convergence of the stationary intensity to the stationary rough Volterra process.

math.PR↗
arXiv2026

Persistent memory and tail-risk amplification in Atlantic Meridional Overturning Circulation variability: a Volterra integral framework calibrated with CMIP6

The Atlantic Meridional Overturning Circulation (AMOC) carries multi-decadal memory that ensemble-mean risk projections do not capture. We quantify this memory and its consequences for tail risk with a Volterra integral framework applied to ensemble-mean-subtracted variability in 14 CMIP6 models (1850-2100, three SSP scenarios), so that results reflect intrinsic thermohaline memory rather than the common anthropogenic forcing trend. Four results follow from leave-one-out cross-validation and an annealed-quenched tail decomposition. First, the ensemble-mean DFA1 Hurst exponent is H = 0.781 +/- 0.219, with 12 of 14 models showing long-range dependence (H > 0.5), consistent with thermohaline adjustment timescales near 33 yr. Second, a first-order Volterra model reduces out-of-sample RMSE by 16.8% over the best autoregressive baseline and by 12.3% over an unconstrained 20-lag distributed baseline, both robust to 200 circular-shift placebos (p < 0.05): the physically motivated kernel shape carries genuine predictive advantage. Third, the rolling 30-year lower-tail frequency rises 1.9-3.1x above the historical baseline depending on scenario, and the memory amplification index exceeds 1 in 8 of 14 models under SSP5-8.5 (median 1.15): persistence-driven clustering of weak-AMOC states amplifies tail frequency beyond forcing-only projections. Fourth, model-specific optimal memory horizons (15-52 yr) correlate with thermohaline regime (r = 0.74, p < 0.01), and the ensemble-mean Hurst exponent already exceeds a detection threshold H* = 0.70 in 9 of 14 models, with theoretical lead times to near-tipping conditions of 10-35 yr depending on scenario. These results support trajectory-specific, memory-aware AMOC risk assessment under moderate-to-high forcing and a physically grounded, scenario-conditional early-warning framework.

physics.ao-ph↗
arXiv2026

Projection, Memory, and the Validity of Scalar Early-Warning Signals: A Schur--Volterra Theory with an AMOC Application

Early-warning signals are usually computed from one or a few observables, although the underlying dynamics are high dimensional. Projection creates a structural ambiguity: the same scalar record may reflect spectral loss of stability, hidden-state memory, nonnormal amplification, a turning point of a physical balance, or a combination of these. We develop a unified operator theory that makes these alternatives explicit. For a resolved-hidden block linearization we derive exact Volterra reductions, finite and diffusive Markovian liftings, and a Schur-resolvent factorization of the stability problem, distinguishing three notions of memory: realized pathwise, structural Schur return, and finite-horizon projected. A covariance theorem gives sufficient conditions for a scalar to inherit critical slowing down from a real multiplier approaching the unit circle: spectral separation, bounded eigenvector conditioning, observability of the critical mode, stochastic excitation of that mode, and quasi-stationary forcing. We then prove failure mechanisms: a critical mode can be invisible or unexcited; an oscillatory crossing need not raise lag-one correlation; a nonnormal family can produce diverging scalar variance with a fixed stable spectrum; a physics-based turning point need not coincide with a spectral boundary; and finite-horizon data cannot identify long memory uniquely. The theory is illustrated with a 4400-year CESM AMOC hysteresis experiment. EOF1 alone detects altered susceptibility, but EOF1-EOF2 is the smallest resolved space revealing changed Schur return and projected memory after recovery; F_ovS reaches its minimum 26 years before collapse while the modal margin does not decrease toward zero: the two diagnostics probe complementary mechanisms. The application is mechanistic and single-trajectory; no validated early-warning signal is claimed.

stat.AP↗
arXiv2026

Memory compression and physical state augmentation favor different AMOC prediction tasks

The Atlantic Meridional Overturning Circulation is monitored and emulated through reduced indices, but such projections discard thermohaline structure and may require either explicit physical state or memory of the observed index. We compare these strategies in 30 branch-consistent CMIP6 trajectories from eight model families using leave-one-family-out validation. Salinity, temperature and density information improves direct 20-year forecasts, whereas compact scalar memory is top-ranked at every recursive horizon and yields the lowest case-averaged Brier score. A matched ablation confirms that feedback from memory improves long-horizon prediction. Physical state and recent trends also predict future ocean-state changes beyond the emissions pathway, most robustly at five years. NorESM under SSP5--8.5 identifies a forcing-dependent limit of scalar compression, while MIROC shows negative long-horizon transfer. A resolvent analysis explains why stable memory components do not guarantee stability of the complete learned model. Physical augmentation and memory compression therefore serve different AMOC prediction tasks.

physics.ao-ph↗
arXiv2026

A Spatial Persistence Gradient in European Warming Consistent with North Atlantic Cold-Blob Influence

Europe is warming faster than the global mean, yet the spatial organisation of this acceleration remains incompletely understood. Using ERA5 reanalysis for 1950--2024 across 28 IPCC AR6 European sub-regions, we identify two connected empirical results. First, the DFA1 Hurst exponent of interannual temperature residuals is strongly and negatively associated with the 1996--2024 warming rate ($r=-0.792$, $p=5.1\times10^{-7}$). High-persistence, mainly Atlantic-proximal regions warm more slowly, whereas low-persistence continental regions warm faster. This relationship is robust to five residualisation schemes, three memory estimators, leave-one-region-out analysis, and five null-test families, including spatial block permutation. It also persists across warming windows ($r=-0.550$ for 1981--2024, $-0.792$ for 1996--2024, and $-0.875$ for 2000--2024), but vanishes under DFA2, indicating that the signal lies in low-frequency interannual-to-decadal persistence rather than trend curvature. The pattern is consistent with, but does not prove, North Atlantic cold-blob and thermohaline influence on European land temperatures. Second, under a strict 2006--2024 holdout, contemporaneous Mediterranean SST reduces mean annual temperature RMSE by 43% (from $0.787$ to $0.449,^{\circ}\mathrm{C}$). A causal lag-weight predictor based on prior-year Mediterranean and Atlantic SST also outperforms AR(2) ($0.578$ versus $0.695,^{\circ}\mathrm{C}$) and remains informative after removing NAO, AO, and PNA effects. Similar skill from five-year moving-average and exponentially weighted predictors shows that short Mediterranean SST persistence at 1--5-year lags is the key predictive ingredient. Together, the results support a two-regime interpretation: Atlantic-proximal regions exhibit stronger memory and oceanic buffering, while continental interiors show faster warming and weaker interannual persistence.

physics.geo-ph↗
arXiv2026

Projected climate memory and inherited warm-tail risk in accelerated European summer warming

European summer warming reflects interactions among background change, persistent ocean--land--circulation states, and same-season variability. We develop an empirical reduced-dynamics framework that decomposes regional summer indicators into inherited slow-state memory, its predictable component, and contemporaneous innovation. Projection-operator theory motivates the decomposition, implemented with finite causal filters, ridge-regularised prediction, and logistic risk models. Using ERA5-derived summer indicators for 28 IPCC AR6 European sub-regions over 1950--2024, with validation on 2006--2024, we find that Mediterranean-state memory improves mean summer-temperature prediction relative to trend-only and ARX baselines. The gain over ARX is modest, while moving-average, exponentially weighted, and tempered filters contain similar annual information, indicating that the data identify useful slow-state memory more robustly than a unique kernel shape. Predictable-state reconstruction is ridge-sensitive and therefore treated diagnostically rather than as a forecasting model. The strongest result concerns warm-tail risk. In parsimonious logistic models, high accumulated Mediterranean memory raises predicted upper-tail event probability by about 8--11 percentage points for annual maximum summer temperature, warm-day frequency, and warm-spell duration at 1-, 3-, and 5-year horizons. Regional bootstrap intervals remain positive for all targets and horizons. Circular-shift placebos yield one-sided probabilities of approximately 0.05--0.14 and do not survive strict family-wise correction across nine tests, so the evidence is moderate rather than decisive. Overall, annual projected memory is not a universal short-horizon predictor, but a physically interpretable inherited risk-loading variable identifying years and regions predisposed to warm-tail outcomes.

physics.ao-ph↗
arXiv2026

Memory operator ensembles indicate proximity to criticality in simulated AMOC transitions

Classical early-warning signals assume that recovery slows near a tipping point, but in a well-studied CESM simulation of AMOC collapse, autocorrelation and variance decrease instead. We develop a preregistered diagnostic that uses only past data to estimate the slowest mode of a memory operator fitted to the evolving overturning structure. The operator ensemble produces an exploratory alarm 77 years before collapse, compared with 29 years for a physics-based freshwater-transport indicator. A blind test on the recovery branch does not validate the same timing rule, supporting interpretation of the diagnostic as a proximity measure rather than a universal clock. In nine additional simulations, it warns before all four finite-rate forced transitions but not before three rapid noise-induced collapses available only as scalar AMOC records. Matched ablations show that the signal depends on long-timescale memory and vertical overturning structure rather than trend, forcing, or multivariate information alone.

stat.AP↗
arXiv2026

Regime-Switching Volterra Operators: Modal Stability and Quenched Amplification

We develop an operator-theoretic framework for finite-dimensional, regime-dependent Volterra equations with completely monotone memory kernels, dissipative network coupling, and Hawkes-type self-excitation. For each fixed regime we construct the associated Volterra resolvent family and prove global well-posedness, continuity across regime switches, and explicit a priori bounds. The main stability result is sharp in the commuting case: after simultaneous diagonalization of the network Laplacian and the excitation operator, each mode obeys a scalar characteristic equation, and global asymptotic stability holds exactly when every modal branching ratio lies below the intensity damping threshold. We also give a norm-based sufficient condition for noncommuting operators and a Perron--Frobenius spectral criterion for nonnegative intensity blocks, showing when norm estimates are conservative. Beyond mean stability, we prove a pathwise finite-range power law for burst amplitudes generated by residence in a Hurwitz but nonnormal regime: under a cone-alignment event, the survival exponent is the ratio of the regime exit rate to a cone-corrected finite-time growth rate bounded above by the logarithmic norm of a fixed Markovian realization in the chosen Euclidean metric. A complementary idealized-feedback result shows how a logarithmic-norm contraction caps the amplification band. Finally, we derive the deterministic intensity block as a mean-field limit of a relaxing long-memory Hawkes system with regimes. Numerical experiments on modal equations, a small-world network, and a switched nonnormal ODE validate the sharp threshold and the finite-range amplification mechanism without using the closed-form tail formula as input.

stat.AP↗
arXiv2026

Intermittency induced by long memory under stochastic regime switching

We study a fundamental instability mechanism in nonlinear, nonlocal dynamical systems arising from the interaction of long-range memory and stochastic regime switching. The dynamics are governed by network-coupled, operator-valued Volterra evolutions with completely monotone memory kernels whose excitation operators and kernel parameters are modulated by an ergodic finite-state continuous-time Markov chain. We formalize a sharp separation between annealed stability (in expectation) and quenched behaviour (along typical sample paths). On the annealed side, we identify an averaged memory gain that yields uniform moment bounds and a memory-adapted Lyapunov functional implying mean-square control under an averaged subcriticality condition. On the quenched side, we show that rare but persistent excursions into supercritical regimes are amplified by memory, producing intermittent macroscopic bursts with heavy-tailed statistics and a deterministic almost sure growth exponent obtained via a subadditive ergodic argument. This establishes an annealed--quenched dichotomy specific to non-Markovian switching systems, where stability in expectation can coexist with pathwise growth and metastable burst phases. We further derive a micro--macro correspondence by proving that a population of regime-modulated self-exciting point processes converges, both annealed and quenched, to the random-coefficient Volterra limit, transferring the burst mechanism from microscopic branching dynamics to macroscopic long-memory flows. Numerical experiments illustrate how burst localization depends on graph geometry and on noncommuting excitation operators.

math.ST↗
arXiv2026

Quenched Amplification and Tail Shaping in Networked Systems with Memory and Regime Switching

Networked systems operating under intermittent adverse conditions and long memory can remain stable on average while exhibiting rare but extreme trajectory-level excursions. We study linear regime-switching network dynamics with Volterra-type memory, formulated through a finite-dimensional lifted ordinary differential equation embedding. Despite finite-horizon annealed boundedness, we show that quenched amplification emerges generically from the interaction of regime persistence, memory accumulation, and non-normal lifted operator geometry. A lower bound on burst-size distributions reveals power-law tails whose exponent is determined by the ratio between unfavorable dwell-time rates and an operator-defined instantaneous growth parameter. This parameter is computable online via the Euclidean logarithmic norm of the lifted operator, yielding a practical early-warning indicator. Building on this structure, we introduce a dynamic data-driven intervention strategy that enforces contraction on demand along rare amplification channels, thereby shaping or truncating tail risk without altering exogenous regime statistics or typical system behavior. The results provide a geometrically grounded and operationally actionable framework for understanding and mitigating extreme events in memory-driven regime-switching systems.

stat.OT↗