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Under mild structural assumptions and regularity conditions on the marginal and conditional densities, an explicit bound on the $β$-mixing coefficients in terms of the physical dependence measure is provided. Consequently, weak physical dependence implies $β$-mixing and strong mixing for triangular arrays, complementing Hill (2025), who proved the converse implication under moment assumptions.

Ordinal pattern dependence has been introduced in order to capture co-monotonic behavior between two time series. This concept has several features one would intuitively demand from a dependence measure. It was believed that ordinal pattern dependence satisfies the axioms which Grothe et al. (Journal of Multivariate Analysis 123, 2014) proclaimed for a multivariate measure of dependence. In the present article we show that this is not true and that there is a mistake in the article Betken et al. (Journal of Multivariate Analysis 186, 2021). Furthermore we show that ordinal pattern dependence satisfies a slightly modified set of axioms.

We introduce the concepts of dependence and independence in a very general framework. We use a concept of rank to study dependence and independence. By means of the rank we identify (total) dependence with inability to create more diversity, and (total) independence with the presence of maximum diversity. We show that our theory of dependence and independence covers a variety of dependence concepts, for example the seemingly unrelated concepts of linear dependence in algebra and dependence of variables in logic.

This note shows that multiplicative Kantian optimization is reference-point dependent. In the standard formulation, the Multiplicative Kantian Equilibrium (MKE) test is applied to absolute contributions, and Nash optimizers may free-ride on Kantians. We compare this with a formulation in which Kantians apply the same multiplicative test to contributions above the individually rational Nash level. With the reference point at the Nash contribution, the Nasher's action is zero in the auxiliary scale. Since zero is a fixed point of every multiplicative transformation, the Kantian test makes the Nash outcome stable in mixed encounters. In a public-good game with a dominant Nash strategy, this neutralizes the free-riding advantage of Nash optimizers, while two Kantians attain the efficient symmetric MKE under the efficient-equilibrium selection convention. The induced type-payoff matrix makes the Kantian type weakly dominant and evolutionarily stable. Thus, the stability of Kantian behavior depends not only on the multiplicative Kantian criterion, but also on the reference point used to formulate it.

The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges, and recent developments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimizatio

We study the dependence control theory, with a focus on the tail property and dependence transformability of wireless channel capacity, respectively, from the perspective of an information theoretic model of the wireless channel and from the perspective of a functional of controllable and uncontrollable random parameter processes. We find that the light-tailed behavior is an intrinsic property of the wireless channel capacity, which is due to the passive nature of the wireless propagation environment and the power limitation in the practical systems. We observe that the manipulation of the marginal distributions has a bias in favor of positive dependence and against negative dependence, e.g., when a parameter process bears negative dependence, the increases of the means of marginals can not leads effectively to a better system performance. On the other hand, the dependence bias indicates that the dependence is a tradable resource, i.e., when the dependence resource is utilized another resource can be saved. For example, the negative dependence can be traded for transmission power in terms of the performance measures.

We introduce a new stochastic order for the tail dependence between random variables. We then study different measures of tail dependence which are monotone in the proposed order, thereby extending various known tail dependence coefficients from the literature. We apply our concepts in an empirical study where we investigate the tail dependence for different pairs of S&P 500 stocks and indices, and illustrate the advantage of our measures of tail dependence over the classical tail dependence coefficient.

In this article, we show that the recently introduced ordinal pattern dependence fits into the axiomatic framework of general multivariate dependence measures, i.e., measures of dependence between two multivariate random objects. Furthermore, we consider multivariate generalizations of established univariate dependence measures like Kendall's $τ$, Spearman's $ρ$ and Pearson's correlation coefficient. Among these, only multivariate Kendall's $τ$ proves to take the dynamical dependence of random vectors stemming from multidimensional time series into account. Consequently, the article focuses on a comparison of ordinal pattern dependence and multivariate Kendall's $τ$ in this context. To this end, limit theorems for multivariate Kendall's $τ$ are established under the assumption of near-epoch dependent data-generating time series. We analyze how ordinal pattern dependence compares to multivariate Kendall's $τ$ and Pearson's correlation coefficient on theoretical grounds. Additionally, a simulation study illustrates differences in the kind of dependencies that are revealed by multivariate Kendall's $τ$ and ordinal pattern dependence.

This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence are discussed: continuous dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games.

We propose a local and general dependence quantifier between two random variables $X$ and $Y$, which we call Local Lift Dependence Scale, that does not assume any form of dependence (e.g., linear) between $X$ and $Y$, and is defined for a large class of random variables, singular and absolutely continuous w.r.t Lebesgue measure. We argue that this dependence scale is more general and suitable to study variable dependence than other specific local dependence quantifiers and global dependence coefficients, as the Mutual Information. An outline of how this dependence scale may be useful in branches of applied probability and topics for future research are presented.

This paper presents a general framework for modeling dependence in multivariate time series. Its fundamental approach relies on decomposing each signal in a system into various frequency components and then studying the dependence properties through these oscillatory activities.The unifying theme across the paper is to explore the strength of dependence and possible lead-lag dynamics through filtering. The proposed framework is capable of representing both linear and non-linear dependencies that could occur instantaneously or after some delay(lagged dependence). Examples for studying dependence between oscillations are illustrated through multichannel electroencephalograms. These examples emphasized that some of the most prominent frequency domain measures such as coherence, partial coherence,and dual-frequency coherence can be derived as special cases under this general framework.This paper also introduces related approaches for modeling dependence through phase-amplitude coupling and causality of (one-sided) filtered signals.

Ordinal pattern dependence is a multivariate dependence measure based on the co-movement of two time series. In strong connection to ordinal time series analysis, the ordinal information is taken into account to derive robust results on the dependence between the two processes. This article deals with ordinal pattern dependence for long-range dependent time series including mixed cases of short- and long-range dependence. We investigate the limit distributions for estimators of ordinal pattern dependence. In doing so we point out the differences that arise for the underlying time series having different dependence structures. Depending on these assumptions, central and non-central limit theorems are proven. The limit distributions for the latter ones can be included in the class of multivariate Rosenblatt processes. Finally, a simulation study is provided to illustrate our theoretical findings.

The target-mass number dependence of nucleon-nucleon pairs with short-range correlations is explored in a physically transparent geometrical model within a zero-range approximation. The observed $A$ dependence of 2-nucleon ejection cross sections in $(e,e')$ reactions is found to reflect the mass dependence of nuclear density distributions. A parametrization of this $A$ dependence is given. The $A$ dependence of proton-proton vs.\ proton-neutron pairs relative to $^{12}$C is also analyzed in this model. It can be understood using simple combinatorics without any additional isospin dependence.

We extend the treatment of functional dependence, the basic concept of dependence logic, to include the possibility of dependence with a limited number of exceptions. We call this approximate dependence. The main result of the paper is a Completeness Theorem for approximate dependence atoms. We point out some problematic features of this which suggests that we should consider multi-teams, not just teams.

We study the complexity of producing $(δ,ε)$-stationary points of Lipschitz objectives which are possibly neither smooth nor convex, using only noisy function evaluations. Recent works proposed several stochastic zero-order algorithms that solve this task, all of which suffer from a dimension-dependence of $Ω(d^{3/2})$ where $d$ is the dimension of the problem, which was conjectured to be optimal. We refute this conjecture by providing a faster algorithm that has complexity $O(dδ^{-1}ε^{-3})$, which is optimal (up to numerical constants) with respect to $d$ and also optimal with respect to the accuracy parameters $δ,ε$, thus solving an open question due to Lin et al. (NeurIPS'22). Moreover, the convergence rate achieved by our algorithm is also optimal for smooth objectives, proving that in the nonconvex stochastic zero-order setting, nonsmooth optimization is as easy as smooth optimization. We provide algorithms that achieve the aforementioned convergence rate in expectation as well as with high probability. Our analysis is based on a simple yet powerful lemma regarding the Goldstein-subdifferential set, which allows utilizing recent advancements in first-order nonsmooth nonconvex

Measuring dependence between two random variables is very important, and critical in many applied areas such as variable selection, brain network analysis. However, we do not know what kind of functional relationship is between two covariates, which requires the dependence measure to be equitable. That is, it gives similar scores to equally noisy relationship of different types. In fact, the dependence score is a continuous random variable taking values in $[0,1]$, thus it is theoretically impossible to give similar scores. In this paper, we introduce a new definition of equitability of a dependence measure, i.e, power-equitable (weak-equitable) and show by simulation that HHG and Copula Dependence Coefficient (CDC) are weak-equitable.

We investigate, by means of numerical lattice simulations, the $θ$-dependence of the critical deconfinement temperature of $\mathrm{SU}(N)$ gauge theories at large $N$: $T_c(θ) = T_c(0)[1-Rθ^2+O(θ^4)]$, with $R\sim O(1/N^2)$. We follow two different strategies to determine $R$, one based on the calculation of the latent heat of the transition and on the jump of the topological susceptibility at the $θ=0$ critical point, the other relying on a direct probe of $T_c(θ)$ by means of imaginary-$θ$ Monte Carlo simulations. Our results show that $R$ follows the expected large-$N$ scaling.

Measuring the dependence of data plays a central role in statistics and machine learning. In this work, we summarize and generalize the main idea of existing information-theoretic dependence measures into a higher-level perspective by the Shearer's inequality. Based on our generalization, we then propose two measures, namely the matrix-based normalized total correlation ($T_α^*$) and the matrix-based normalized dual total correlation ($D_α^*$), to quantify the dependence of multiple variables in arbitrary dimensional space, without explicit estimation of the underlying data distributions. We show that our measures are differentiable and statistically more powerful than prevalent ones. We also show the impact of our measures in four different machine learning problems, namely the gene regulatory network inference, the robust machine learning under covariate shift and non-Gaussian noises, the subspace outlier detection, and the understanding of the learning dynamics of convolutional neural networks (CNNs), to demonstrate their utilities, advantages, as well as implications to those problems. Code of our dependence measure is available at: https://bit.ly/AAAI-dependence

We investigate the relative information content of six measures of dependence between two random variables $X$ and $Y$ for large or extreme events for several models of interest for financial time series. The six measures of dependence are respectively the linear correlation $ρ^+_v$ and Spearman's rho $ρ_s(v)$ conditioned on signed exceedance of one variable above the threshold $v$, or on both variables ($ρ_u$), the linear correlation $ρ^s_v$ conditioned on absolute value exceedance (or large volatility) of one variable, the so-called asymptotic tail-dependence $λ$ and a probability-weighted tail dependence coefficient ${\bar λ}$. The models are the bivariate Gaussian distribution, the bivariate Student's distribution, and the factor model for various distributions of the factor. We offer explicit analytical formulas as well as numerical estimations for these six measures of dependence in the limit where $v$ and $u$ go to infinity. This provides a quantitative proof that conditioning on exceedance leads to conditional correlation coefficients that may be very different from the unconditional correlation and gives a straightforward mechanism for fluctuations or changes of correlatio