Functional dependence measures have become an important tool in the analysis of nonlinear time series and are typically formulated with respect to a given innovation representation of the process. This note points out that the probability space on which such representations yield the expected memory loss properties may not always coincide with the natural dynamical probability space of the model. We exhibit classes of uniformly ergodic autoregressive processes for which the behavior of the natural innovation coupling undergoes a qualitative transition as the model parameter varies. For this family of models, this transition coincides with a change in the sign of an associated Lyapunov exponent. In particular, a positive Lyapunov exponent may prevent the forgetting of initial perturbations along trajectories driven by the same innovations, despite uniform ergodicity of the associated Markov chain. These observations highlight the importance of carefully specifying the underlying probability space when interpreting or applying functional dependence measures.
We consider simple autoregressive processes of type AR(1), whose stationary distribution is supported on a subset of the unit interval and is an affine transformation of a Bernoulli convolution. A new structural representation of the stationary distribution as a product of a power function with a log-periodic function near the origin is given, which gives structural insight to the stationary distribution on the whole unit interval by using a characteristic functional equation. This enables to prove that the stationary distribution of the AR(1)-process belongs to the domain of geometric partial attraction of a max-semistable law. We further prove uniform convergence of the distribution function of normalized maxima of the AR(1)-process to a certain power of the max-semistable law in the spirit of a merge theorem and point out connections to deterministic and random dynamical systems.
The Functional Autoregressive Model (FAR) generalizes the multivariate AR(1) model in Time Series Analysis to functional data. It serves as a historical foundational point in the study of functional time series and remains a fundamental and widely used model for dependent functional data. The process-the observed data-generated by the FAR model forms a Hilbert-valued Markov chain. This paper investigates the non-asymptotic prediction mean square error and derives a lower bound. This lower bound is established in the specific context of non-i.i.d. data and depends on the mixed smoothness of the functional time series and of the unknown correlation operator driving the FAR model. Instead of the standard functional PCA regularization, a ridge-type estimator is proposed, which avoids the preliminary estimation of the spectrum of the covariance sequence associated with the process. A non-asymptotic upper bound is derived for this estimate, which matches the lower bound up to multiplicative constants. Furthermore, a detailed study of the estimate's bias reveals connections between functional smoothness parameters and regularly/rapidly varying functions, which are common in extreme value theory. Simulation results corroborate the theoretical main theorems.
We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system. More precisely, we study discrete-time autonomous stochastic systems defining time-homogeneous Markov processes. We first focus on estimating the optimal one-step prediction function by nonlinear least squares, and derive high-probability guarantees measured with respect to the invariant measure of the process. These results make explicit how the non-independent and non-identically distributed nature of trajectory data modifies the classical statistical learning analysis. We then extend the framework to higher-order systems and finite-state spaces. Finally, we show that the same least squares and concentration arguments naturally extend to learning Koopman operators. Our approach combines tools from statistical learning theory and quantitative ergodic theory for Markov chains. It relies, in particular, on a concentration inequality for Hilbert-space-valued additive functionals of uniformly geometrically ergodic Markov chains.
Oleksii Kachaiev, S. Villa, Lorenzo Rosasco· 0 citations
We construct early-warning signals for impending critical transitions in non-Markovian systems. We analyze stochastic forcings such as fractional Brownian motion, fractional Ornstein-Uhlenbeck processes and red noise in fast-slow systems exhibiting such transitions. We show that the effectiveness of indicators such as autocovariance, autocorrelation, and spectral density depends on several properties of the underlying system. In particular, we compare the influence of the Hurst index and the bifurcation type. We prove that the rotatory dynamics associated with a Hopf bifurcation substantially alters the scaling laws of these observables. Finally, we provide practical guidelines for implementing these signals and validate them on both theoretical and applied models.
Paolo Bernuzzi, Alexandra Blessing, Dennis Rudik· 0 citations
We introduce a self-exciting point process with power-law intensity dynamics that admits a finite-dimensional Markovian state representation. The model is constructed to preserve the local jump and slope update structure of power-law Hawkes processes, while replacing global history dependence with a nonlinear Markov chain governing the intensity dynamics. Within a general state-space framework, we establish irreducibility, aperiodicity, and the T-chain property under mild regularity conditions on the inter-arrival time distribution. Under an explicit stability condition, we further prove that the latent state process is positive Harris recurrent, ensuring the existence of a unique invariant distribution. Simulation results based on the local Whittle estimator show that the proposed Markovian intensity model exhibits long-memory behavior near the boundary of the stability region.
Empirical entropy rates are widely used to quantify unpredictability from symbolic or time-series data, yet their interpretation is subtle in weakly chaotic dynamics, where ordinary Lyapunov exponents vanish and invariant measures are infinite. We address this issue by studying the empirical one-step conditional entropy for the fixed finite partitions considered below in one-dimensional intermittent maps with infinite invariant measures. For the modified Bernoulli map and the Boole transformation in the infinite-measure weak-chaos regime, we prove that this per-step empirical entropy converges to zero. Thus, the usual entropy-rate normalization becomes asymptotically blind to subexponential instability. The finite-time information sum, however, remains informative. Rare transitions between long laminar phases occur on the return-sequence scale, and their empirical self-information contributes an additional logarithmic factor. Under the stated regularity and moment assumptions, this mechanism yields a two-term estimate for the ensemble mean decay, supported by numerical simulations. Although the raw entropy rate vanishes, self-normalized fluctuations remain nontrivial and are numerically consistent with normalized Mittag-Leffler laws. A comparison with generalized Lyapunov sums shows that the corresponding information sum is not a Krengel entropy estimator, but a computable, partition-dependent finite-time measure of sparse symbolic transitions. These results clarify what empirical Markov entropy can, and cannot, measure in infinite-measure weak chaos.
