mixing time estimation

This paper addresses the problem of estimating the stationary distribution—i.e., the probability mass vector over states—of an α-mixing stochastic process from a single trajectory of length *n*, using empirical state frequencies. The estimation error is measured in total variation distance. Methodologically, we extend the WingIt estimator to α-mixing processes for the first time; propose a novel hybrid strategy combining plug-in estimation with WingIt; and derive a self-normalized concentration inequality tailored to mixing sequences, circumventing the failure of Poissonization under non-i.i.d. dependence. Theoretically, our estimator achieves universal consistency as *n* → ∞ for arbitrary finite state spaces and general α-mixing processes. It recovers existing i.i.d. results in the degenerate case and provides the first frequency-to-mass estimation framework for Markov and broader dependent processes with rigorous theoretical guarantees.

This paper addresses the weak statistical foundations of nonlinear time series models in econometrics, queueing theory, and machine learning, specifically focusing on (i) how α-mixing of exogenous regressors propagates to the response variable, and (ii) mixing preservation and limiting behavior of nonstationary Markov chains under stochastic nonstationarity. We develop a rigorous propagation mechanism based on first-passage coupling and establish an analytical framework for nonstationary Markov chains incorporating drift conditions and small-set structures. Departing from conventional stationarity assumptions, we derive, for the first time in nonstationary stochastic environments, a strong law of large numbers (LLN) and a functional central limit theorem (FCLT) for weakly dependent nonlinear sequences. Our results yield novel theoretical foundations for asymptotic normality and provide a stability criterion with guaranteed mixing properties for single-server queueing systems.

Non-asymptotic mixing-time analysis of finite-state ergodic Markov chains—both reversible and irreversible—remains challenging, particularly due to the lack of a unified convergence characterization for irreversible chains. Method: We develop an operator-theoretic framework based on orthogonal projections of the transition operator in the ℓ²(π) space, using matrix norms to quantify convergence rates. Contribution/Results: We establish, for the first time, submultiplicativity of pointwise χ²-divergence for irreversible chains, yielding explicit, computable bounds dependent on spectral structure, algebraic–geometric multiplicity gaps, and condition numbers of similarity transformations. We apply this framework to momentum-based samplers, revealing their diffusive behavior via hypercontractivity and regression analysis. For irreversible triangular random walks on graphs, we derive the tightest known mixing-time bound to date and demonstrate near-optimal O(n¹·⁹⁷) convergence even under V-shaped target distributions.

This paper addresses the estimation of β-mixing coefficients for geometrically ergodic Markov processes, aiming to quantify temporal dependence from a single observed trajectory. We propose a nonparametric estimation framework grounded in kernel density estimation and statistical learning theory, integrating tools from Besov space analysis, mixing process theory, and concentration inequalities. Our main contributions are threefold: (1) Under Besov-type regularity assumptions on the stationary density, we establish the first convergence rate with an explicit logarithmic correction factor; (2) For finite-state-space Markov chains, we achieve the optimal expected error rate $O(log n / sqrt{n})$ and corresponding high-probability bounds—without requiring any density boundedness or smoothness assumptions; (3) In continuous state spaces, we derive an expected error rate of $O(log n cdot n^{-s/(2s+2)})$, where $s > 0$ denotes the Besov smoothness index, along with matching high-probability guarantees.

This paper addresses the challenge of efficiently and analytically modeling process execution time statistics from event logs. Methodologically, it introduces the first end-to-end analytical performance analysis framework based on semi-Markov processes: it directly infers execution time means and probability density functions (PDFs) from logs—bypassing simulation entirely. For discrete-time execution times, it employs exact convolution; for continuous-time cases, it approximates PDFs using Gaussian mixture models (GMMs), balancing accuracy, model compactness, and interpretability. Experiments show that the discrete-time approach achieves up to one order of magnitude speedup over simulation under small support sets, while GMM-based representation drastically reduces model size, enabling rapid what-if analysis. The core contribution is the first fully analytical, log-driven inference of semi-Markov performance models—eliminating reliance on traditional simulation-based approaches and establishing a new paradigm for scalable, interpretable process performance analysis.

This work addresses the lack of a unified computational framework for analyzing non-ergodicity, modeling heavy-tailed dynamics, and studying decision-making under uncertainty in stochastic processes. To this end, we introduce an open-source Python library that, for the first time, integrates non-ergodicity diagnostics, simulation of heavy-tailed processes—such as multiplicative Lévy growth and memory-dependent mean-reverting dynamics—and agent-based experimentation within a single platform. Built upon the scientific Python ecosystem (NumPy/SciPy), the library supports end-to-end workflows including stochastic process definition, simulation, parameter inference, and partial solution of stochastic differential equations. Through several reproducible examples—ranging from heavy-tailed ensemble diffusion to pre-asymptotic fluctuation analysis—it substantially reduces boilerplate code and enhances both reproducibility and development efficiency in the study of time-averaged behaviors of complex stochastic systems.

This work addresses the problem of one-sided α-correct sequential hypothesis testing for data generated by ergodic Markov chains, where the null and alternative hypotheses correspond to disjoint sets of transition matrices. For the first time, it jointly incorporates the stationary distribution and transition structure into a non-asymptotic lower bound analysis, establishing a tight instance-dependent lower bound. Leveraging information-theoretic tools, ergodic theory of Markov chains, and sequential testing design, the paper proposes an optimal test whose expected stopping time asymptotically achieves this bound as α → 0. This approach overcomes the limitations of existing methods that are either only asymptotically optimal or suboptimal, providing a sharp characterization of sequential testing for Markovian data. The framework is successfully applied to detecting misspecification in MCMC models and sequentially verifying linear structural assumptions on transition dynamics in Markov decision processes.

This paper addresses the challenge of parameter estimation for stable continuous-state branching processes (CSBPs) under partial observation. We propose a novel inference framework grounded in the subordinator representation of CSBPs. Our core innovation lies in fully mapping the stochastic dynamics of CSBPs into the subordinator domain, thereby circumventing reliance on closed-form transition densities. Specifically, we reconstruct the likelihood function assumption-free via Laplace transforms and their numerical inversion, while simultaneously developing a differentiable discrete-time trajectory simulator. The method achieves statistical consistency and computational feasibility for stable CSBPs, markedly improving both estimation accuracy and efficiency. To our knowledge, this is the first approach enabling end-to-end parameter inference and simulation within the subordinator domain.

This work addresses the challenge of causal inference with continuous-time marked point process data, for which existing methods lack a suitable identification framework. Building on martingale theory, the authors extend the core assumptions of discrete-time causal inference—consistency, exchangeability, and positivity—to the continuous-time setting. They formulate a dynamic treatment strategy and a potential outcomes model tailored to marked point processes and establish corresponding causal identification conditions. Leveraging this foundation, they derive a novel marginal g-formula that enables nonparametric identification of causal effects. The proposed framework subsumes existing results for discrete-time and counting process settings as special cases, demonstrating both theoretical compatibility and extensibility, thereby unifying survival analysis and causal inference within a coherent paradigm.