Real-life financial time series exhibit heavy tails and clusters of extreme values. In this talk we will address models that exhibit these stylized facts. This is the class of regularly varying time series, introduced by Davis and Hsing (1995, AoP) and further developed by Basrak and Segers (2009, SPA). The marginal distribution of a regularly varying time series has tails of power-law type, and the dynamics caused by an extreme event in this time series is described by the spectral tail process. The perhaps best known financial time series models of this kind are Engle’s (1982) ARCH process, Bollerslev’s (1986) GARCH process and Engle’s and Russell’s (1998) Autoregressive Conditional Duration (ACD) model. The length and magnitude of extremal clusters in such a series can be described by an analog of the autocorrelation function for extreme events: the extremogram. The extremal index is another useful tool for describing expected extremal cluster sizes. Both objects can be expressed in terms of the spectral tail process and allow for statistical estimation. The probabilistic and statistical aspects of regularly varying time series are summarized in the recent monograph by Mikosch and Wintenberger (2024) “Extreme Value Theory for Time Series. Models with Power-Law Tails”. The talk is based on joint work with Olivier Wintenberger (Sorbonne).

Consider the regression problem where the response and the covariate are unmatched. Under this scenario, we do not have access to pairs of observations from their joint distribution, but instead we have separate data sets of responses and covariates, possibly collected from different sources. We study this problem assuming that the regression function is linear and the noise distribution is known or can be estimated. We introduce an estimator of the regression vector based on deconvolution (the DLSE) and demonstrate its consistency and asymptotic normality under parametric identifiability. Under non-identifiability of the regression vector but identifiability of the distribution of the predictor, we construct an estimator of the latter based on the DLSE and show that it converges to the true distribution of the predictor at the parametric rate in the Wasserstein distance of order 1. We illustrate the theory with several simulation results. \[ \] This talk is based on my joint work with Mona Azadkia, Antonio di Noia and Cecile Durot

We consider the task of learning dynamical systems from data in a system-agnostic framework. This presentation is divided into two parts. First, we utilize a simulation-based approach to investigate algorithms for learning chaotic Ordinary Differential Equations (ODEs). We demonstrate that for noise-free data and low-dimensional systems, this task is effectively solved, as polynomial regression-based methods can achieve machine-precision forecasts. However, we show that observational noise remains a significant challenge for most algorithms. In the second part, we address this challenge by developing nonparametric statistical theory for learning ODEs from noisy observations. Specifically, we establish minimax optimal error rates for two contrasting observational models: the Stubble model, consisting of many short trajectories, and the Snake model, consisting of a single long trajectory. We conclude by discussing challenges at the intersection of dynamical systems and statistical learning.

Suppose we are given some data, and we hypothesize a structural causal model to describe them: how can we narrow the set of causal graphs compatible with our observations? The theory of identifiability aims to answer this question. We show that, in the case of additive noise models, the score function of the data contains all the information about the causal graph. However, this requires strong and, crucially, hard-to-verify modeling assumptions, like additivity of the noise. When direct experiments to infer causality are not feasible, this raises the question: how can we move past these restrictions? Borrowing ideas from independent component analysis, we show how multiple environments (read: non i.i.d. data) can overcome these limitations: for structural causal models with arbitrary causal mechanisms, data from only three environments uniquely identify the causal graph from the Jacobian of the score function. Thus, non-i.i.d.-ness turns from a curse into a blessing for causal discovery.

ICA-based causal discovery methods such as LiNGAM have been highly successful under the assumption that noise variables become independent after an appropriate causal ordering. However, this assumption is often violated in the presence of confounding. \[ \] In this talk, I present a Bayesian and information-theoretic formulation of ICA for causal order estimation that explicitly allows for confounding. Rather than enforcing independence, we quantify residual dependence among noise variables using multivariate mutual information and evaluate causal orders via Bayesian marginal likelihoods. \[ \] This approach provides a principled ranking of causal orders under confounding and recovers classical LiNGAM-type methods as special cases when confounding is absent. I will focus on the conceptual framework and discuss connections to existing ICA-based methods, as well as open questions.

Mercer's celebrated theorem is refined and extended by introducing a novel class of higher-order kernel operators that includes the common integral operator only as a special case.

These operators genuinely take into account information encoded in the (weak) derivatives of a kernel, and their natural domains are Sobolev spaces of order k over some bounded d-dimensional space. domain, where k depends on the order of (weak) differentiability.

The spectral decomposition of such higher-order kernel operators leads to Mercer-type expansions, which are optimal in terms of the Sobolev norm and, if k>d, also converge uniformly without requiring the kernel to be positive definite.

Nuclearity of higher order kernel operators is confirmed for positive definite kernels, and a major refinement of Mercer's theorem is obtained that implies trace formulas and a simple rate for the uniform convergence (including derivatives) in terms of the eigenvalues. A further immediate consequence is novel spectral representations of RKHS's.

Finally, applied to the covariance kernel of a (weakly) differentiable stochastic process, these refinements also yield novel Karhunen-Loève-type expansions allowing for simultaneous approximations of the process and its (weak) derivatives in a mean-square-optimal sense.

Contemporary data analysis pipelines often involve the use and reuse of data. For instance, a scientist may explore a dataset to select an interesting hypothesis, and then wish to test this hypothesis with the same data. From a statistical perspective, this double use of data is highly problematic: it induces dependence between the hypothesis generation and testing stages, which complicates inference. Failure to account for this dependence renders classical inference techniques invalid.

I will present "data thinning", a set of strategies for obtaining independent training and test sets so that the former can be used to select a hypothesis, and the latter to test it. Data thinning enables valid selective inference in settings for which no solutions were previously available. However, it is also restrictive, in the sense that it requires strong distributional assumptions. Therefore, I will also present two strategies inspired by data thinning that enable valid post-selection inference without such assumptions. One strategy considers thinning summary statistics of the data, rather than the data itself, in order to take advantage of asymptotic properties of the summary statistics. The second strategy involves generating training and test sets that are not independent, and then orthogonalizing the latter with respect to the former in order to conduct valid inference.

In network science, as in other applied fields, answering key questions often requires distinguishing causal mechanisms from context-dependent associations. One formalization of causality uses invariance: a relationship between a target variable and a set of covariates is causal if it remains stable across diverse environments or interventions. Originally developed for linear models, this approach has been extended to more flexible frameworks, including generalized linear models, where causality is expressed via Pearson risk invariance.

In this talk, we extend causal invariance to dynamic networks, focusing on Relational Event Models (REMs), which describe instantaneous interactions over time between social actors. Because the target —the dynamic network— and potential causal drivers are stochastic processes, dependences among them are characterized using conditional local independence. By exploiting a connection between REMs and logistic regression, we extend Pearson risk invariance to this dynamic setting, producing a causal discovery algorithm that only requires data from a single observational environment. We show how this approach is able to identify, and distinguish, important causal mechanisms in network science, such as social influence and homophily.

This is joint work with Melania Lembo and Ernst Wit (USI).

Conditional-independence-based discovery uses statistical tests to identify a graphical model that represents the independence structure of variables in a dataset. These tests, however, can be unreliable, and algorithms are sensitive to errors and violated assumptions. Often, there are tests that were not used in the construction of the graph. In this talk, we will see that these redundant tests have the potential to detect or sometimes correct errors in the learned model. But we will further see that not all tests contain this additional information and that such redundant tests have to be applied with care. Precisely, we argue that the conditional (in)dependence statements that hold for every probability distribution are unlikely to detect and correct errors - in contrast to those that follow only from graphical assumptions.

Generative modeling builds on and substantially extends the classical idea of generating synthetic data from observed samples. In this talk, I will show that this principle provides a natural and theoretically well-founded foundation for bootstrap inference. The resulting method yields statistically valid confidence intervals for both regular and irregular estimators, including settings in which Efron’s bootstrap fails. From this perspective, the generative-modeling bootstrap could be viewed as a modern extension of the smoothed bootstrap: it has the potential to mitigate the curse of dimensionality and remain effective in challenging regimes where estimators may not be root-n consistent or admit a Gaussian limit.

The quantity of interest in the classical Cramér–Rao theory of unbiased estimation (i.e., the Cramér–Rao lower bound, exact efficiency in exponential families, and asymptotic efficiency of maximum likelihood estimation) is the variance, which represents the instability of an estimator when its value is compared to the value for an independently sampled data set from the same distribution. In this paper, we study a different quantity that captures the instability of an estimator when its value is compared to that obtained under an infinitesimal additive perturbation of the original data set; we refer to this as the sensitivity of an estimator.

The resulting theory of sensitivity is based on Wasserstein geometry in much the same way that the classical theory of variance is based on Fisher–Rao (equivalently, Hellinger) geometry. This perspective yields several results paralleling the classical case: a Wasserstein–Cramér–Rao lower bound for the sensitivity of any unbiased estimator, a characterization of models admitting unbiased estimators that attain this bound exactly, and a guarantee that Wasserstein projection estimators achieve the bound asymptotically. We illustrate the theory through a range of statistical examples, in some cases revealing new optimality properties of existing estimators and in others introducing new ones.

This is joint work with Nicolas Garcia Trillos (University of Wisconsin) and Adam Jaffe (Columbia), based on the paper: https://arxiv.org/pdf/2511.07414.

Continuous-time stochastic systems are often used to model causal relationships between variables that evolve over time. In many applications, however, the full time evolution is not available. Instead, we only observe the system after it has reached stationarity. This leads to the question of which causal information can be recovered from such stationary data.

In this talk, I will consider this question for linear stochastic differential equations when the causal structure is known. I will explain why identifying the exact strength of a direct causal effect is generally not possible without additional assumptions. Instead, I will focus on whether a direct causal effect is positive or negative. This leads to the notion of edge-sign identifiability, which asks when the sign of a direct causal effect is uniquely determined by the stationary covariance matrix. I will present criteria for determining whether an edge sign is identifiable, non-identifiable, or partially identifiable, and illustrate these criteria using examples from both classical causal structures and cyclic systems.

Functional data analysis (FDA) provides statistical methods for analyzing samples of time-continuous stochastic processes. Measurements often arise in the form of sensor data for a scientific variable. The practical problem of irregular sensor disruptions has fostered interest in analyzing partially observed random functions. To allow statistical analysis, we develop the first inference methods for dependent, partially observed functional time series. Mathematically, we model data on the space of bounded functions equipped with the supremum norm. This allows simultaneous inference across the entire functional domain, including simultaneous confidence bands -- something existing Hilbert-space-based methods cannot provide. To study non-stationary trends along the time series, we extend state-of-the-art multiscale inference methods (originally developed for scalar data) to partially observed functions. The key application of the latter methods is testing for excessive pollution levels in inner cities. Interestingly, our results also improve on existing results for fully observed functional time series by avoiding a functional CLT.

We present FLOP (Fast Learning of Order and Parents), a score-based causal discovery algorithm for linear models. It pairs fast parent selection with iterative Cholesky-based score updates, cutting run-times over prior algorithms. This makes it feasible to fully embrace discrete search, enabling iterated local search with principled order initialization to find graphs with scores at or close to the global optimum. The resulting structures are highly accurate across benchmarks, with near-perfect recovery in standard settings. This performance calls for revisiting discrete search over graphs as a reasonable approach to causal discovery.

Causal reasoning is usually formalized through structural causal models (SCMs) or potential outcomes. These frameworks have been enormously successful for modeling, identification, and inference, but they are not primarily designed as axiomatic foundations analogous to probability spaces in probability theory. This tutorial introduces causal spaces, a measure-theoretic framework in which interventions are represented by primitive causal kernels satisfying two minimal axioms: doing nothing leaves everything unchanged, and intervened coordinates take their prescribed values. The tutorial will explain the motivation for causal spaces, present the basic definition and semantics, and work through examples linking back to familiar causal models. It will then present some further development of basic causal space theory, such as causal effects, sources, identifiability, and counterfactual spaces, and touch upon advanced topics such as targeted interventions and continuous-time stochastic processes that are more difficult to express in existing frameworks.

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