stationary distribution analysis

This paper addresses risk assessment of rare events in nonstationary complex systems, focusing on modeling heavy-tailed multivariate distributions of interdependent variables and elucidating how time-varying dependence amplifies tail risk. We propose a novel class of random matrix models that unifies Gaussian and algebraic heavy-tailed characteristics, deriving—for the first time—closed-form joint distributions and explicit moment expressions. In the algebraic case, the model reduces the number of fitting parameters by one to two, substantially enhancing practicality. The methodology integrates scalar products of generalized correlation matrices, random matrix theory, heavy-tailed distribution modeling, and analytical derivation of joint distributions for linear combinations. Empirical validation on financial data demonstrates high accuracy. The framework provides an interpretable, computationally tractable theoretical foundation for extreme-risk quantification and empirical financial market analysis.

This paper addresses asymptotic inference for a single large network sample generated by strategic interaction and homophily among agents in large-scale static and dynamic network formation models. To overcome the challenge of verifying conventional central limit theorems (CLTs) under network moment dependence, we adapt the exponential stabilization condition from stochastic geometry to network analysis—augmented by branching process theory—to derive verifiable primitive sufficient conditions. The resulting CLT framework requires no repeated sampling and applies directly to a single large network. It substantially broadens the theoretical foundation for network parameter estimation and hypothesis testing, and provides the first asymptotic normality guarantee for strategic network models with explicit, quantifiable regularity conditions.

This paper addresses the challenges of modeling multivariate joint distributions and accurately assessing tail risk in nonstationary complex systems—such as financial markets—where conventional methods struggle with time-varying dependence and heavy-tailed dynamics. We propose a parsimonious stochastic matrix model that jointly captures the evolution of time-varying correlations and heavy-tailed structures. Leveraging intraday return data from 479 S&P 500 constituents in 2014, we pioneer the integration of random matrix theory with nonstationary time series analysis and heavy-tailed statistical inference. Empirical results demonstrate that nonstationarity intensifies algebraic tail behavior. The model characterizes the dynamic evolution of the multivariate density function across multiple time scales using a minimal set of interpretable parameters, faithfully reproducing tail thickening induced by correlation drift. It significantly improves the accuracy of extreme-risk measurement for multi-asset portfolios.

Traditional network evolution models focus exclusively on growth, failing to capture the coexistence of network expansion and contraction observed in real-world systems. Method: We propose the first bidirectional dynamic evolution framework that jointly models both degree increase and decrease. Our approach introduces a queueing system to characterize degree evolution, incorporates network shrinkage mechanisms directly into the evolutionary model for the first time, and designs a novel rate-of-degree-change–based preferential attachment mechanism. Contribution/Results: Theoretical analysis and Monte Carlo simulations demonstrate that the model accurately reproduces two canonical degree distributions—long-tailed and parameter-sensitive types. Empirical validation across multiple real-world networks confirms its high predictive accuracy for degree distributions and strong explanatory power for structural dynamics. This work breaks the unidirectional growth paradigm, establishing a unified evolutionary theory capable of explaining both growth-dominated and contraction-coexisting regimes, while revealing a new degree-change-rate–driven preferential attachment mechanism.

This paper investigates the asymptotic behavior of the minimum positive entry—i.e., the smallest positive stationary probability—of the stationary distribution of simple random walk on the directed configuration model with bounded degrees. For sparse random directed graphs with minimum out-degree at least 2, we establish that this minimum decays as $n^{-(1+C+o(1))}$ with high probability, where the exponent $C>0$ is uniquely determined by the competition between the survival probability of a subcritical branching process and the large deviation rate function. This constitutes the first precise exponential characterization of the extremal stationary probability on sparse directed graphs. Moreover, we derive, for the first time, the exact asymptotic orders $n^{1+C+o(1)}$ for both the hitting time and the cover time, revealing an intrinsic scale consistency among these three quantities. Our approach integrates the directed configuration model construction, local weak convergence to trees, branching process analysis, and large deviation theory.

This study addresses the challenge of jointly modeling calibration and control parameters in computer model calibration, where the distribution of calibration parameters is unknown while that of control parameters is known. To tackle this issue, the authors propose a nonparametric Bayesian calibration method based on measure decomposition. The approach preserves the known marginal distribution of the control parameters while employing stochastic process modeling and Bayesian inference to construct a posterior distribution over the input space that aligns with field observations. Notably, this work is the first within a nonparametric calibration framework to explicitly maintain the prior distributional properties of the control parameters, thereby substantially enhancing the physical consistency and scientific credibility of the calibration results.

This work proposes a unified mathematical framework grounded in dynamic information flow for constructing structurally rigorous models of future prediction. By integrating filtering theory, regular conditional probabilities, Markov semigroups, infinitesimal generators, and multiple information geometries—including Hilbert, Fisher–Rao, and Wasserstein—the approach conceptualizes prediction as the construction of conditional distributions governed by informational, geometric, and modeling constraints. The framework elucidates deep connections among classical results such as the tower property and semigroup laws, as well as Itô’s formula and backward equations. Explicit transition laws, spectral decompositions, term structures, and asymptotic behaviors are derived within canonical models like Ornstein–Uhlenbeck and Cox–Ingersoll–Ross, thereby establishing a compact mathematical mapping from idealized theoretical constructs to empirical forecasting.

This work addresses the challenge of causal inference in equilibrium systems confounded by latent variables, where interpretable graphical modeling approaches have been lacking. It introduces antorial graphs into the causal modeling of equilibrium systems for the first time, integrating counterfactual graphs with the Single-World Intervention Graph (SWIG) framework to construct a unified, interpretable causal diagram that jointly represents both observed and counterfactual variables. The proposed method not only yields a clear graphical representation of confounded equilibrium systems but also enables on-demand construction of covariate adjustment sets, offering element-wise flexibility to selectively include or exclude specific variables. This fine-grained control facilitates valid and efficient identification of causal effects under complex equilibrium conditions.

This work proposes a novel small-sample approximation method for Markov chain Monte Carlo (MCMC) simulation that dramatically improves sampling efficiency while preserving the target stationary distribution. Conventional MCMC approaches require millions of sample paths to adequately approximate the stationary distribution, incurring substantial computational costs. In contrast, the proposed technique leverages eigenvalue decomposition to reduce the number of required simulation paths to as few as ten, without compromising distributional fidelity. By integrating Wasserstein distance–based evaluation with explicit modeling of the stationary distribution, the method achieves accuracy comparable to traditional MCMC while significantly reducing estimator variance. This advancement enhances both the stability and computational tractability of MCMC-based inference, offering a scalable alternative for applications where large-scale path simulation is prohibitive.