Academic literature on the topic 'Subadditive sequence'

The topological pressure is defined for subadditive sequence of potentials in bundle random dynamical systems. A variational principle for the topological pressure is set up in a very weak condition. The result may have some applications in the study of multifractal analysis for random version of nonconformal dynamical systems.

For a non-autonomous dynamics with discrete time defined by a tempered sequence of upper-triangular matrices, we obtain lower and upper bounds for the Grobman regularity coefficients. We also give two applications of these results: we obtain an upper bound for the Grobman coefficients of the exterior powers of a tempered sequence, and we give a simple proof of Oseledets' multiplicative ergodic theorem for cocycles over a measure-preserving transformation without using Kingman's subadditive ergodic theorem.

The infinite combinatorics here give statements in which, from some sequence, an infinite subsequence will satisfy some condition - for example, belong to some specified set. Our results give such statements generically - that is, for 'nearly all' points, or as we shall say, for quasi all points - all off a null set in the measure case, or all off a meagre set in the category case. The prototypical result here goes back to Kestelman in 1947 and to Borwein and Ditor in the measure case, and can be extended to the category case also. Our main result is what we call the Category Embedding Theorem, which contains the Kestelman-Borwein-Ditor Theorem as a special case. Our main contribution is to obtain function wise rather than point wise versions of such results. We thus subsume results in a number of recent and related areas, concerning e.g., additive, subadditive, convex and regularly varying functions.

Enhancers regulate gene expression through the binding of sequence-specific transcription factors (TFs) to cognate motifs. Various features influence TF binding and enhancer function—including the chromatin state of the genomic locus, the affinities of the binding site, the activity of the bound TFs, and interactions among TFs. However, the precise nature and relative contributions of these features remain unclear. Here, we used massively parallel reporter assays (MPRAs) involving 32,115 natural and synthetic enhancers, together with high-throughput in vivo binding assays, to systematically dissect the contribution of each of these features to the binding and activity of genomic regulatory elements that contain motifs for PPARγ, a TF that serves as a key regulator of adipogenesis. We show that distinct sets of features govern PPARγ binding vs. enhancer activity. PPARγ binding is largely governed by the affinity of the specific motif site and higher-order features of the larger genomic locus, such as chromatin accessibility. In contrast, the enhancer activity of PPARγ binding sites depends on varying contributions from dozens of TFs in the immediate vicinity, including interactions between combinations of these TFs. Different pairs of motifs follow different interaction rules, including subadditive, additive, and superadditive interactions among specific classes of TFs, with both spatially constrained and flexible grammars. Our results provide a paradigm for the systematic characterization of the genomic features underlying regulatory elements, applicable to the design of synthetic regulatory elements or the interpretation of human genetic variation.

Abstract For a subshift $(X, \sigma _{X})$ and a subadditive sequence ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ on X, we study equivalent conditions for the existence of $h\in C(X)$ such that $\lim _{n\rightarrow \infty }(1/{n})\int \log f_{n}\, d\kern-1pt\mu =\int h \,d\kern-1pt\mu $ for every invariant measure $\mu $ on X. For this purpose, we first we study necessary and sufficient conditions for ${\mathcal F}$ to be an asymptotically additive sequence in terms of certain properties for periodic points. For a factor map $\pi : X\rightarrow Y$ , where $(X, \sigma _{X})$ is an irreducible shift of finite type and $(Y, \sigma _{Y})$ is a subshift, applying our results and the results obtained by Cuneo [Additive, almost additive and asymptotically additive potential sequences are equivalent. Comm. Math. Phys.37 (3) (2020), 2579–2595] on asymptotically additive sequences, we study the existence of h with regard to a subadditive sequence associated to a relative pressure function. This leads to a characterization of the existence of a certain type of continuous compensation function for a factor map between subshifts. As an application, we study the projection $\pi \mu $ of an invariant weak Gibbs measure $\mu $ for a continuous function on an irreducible shift of finite type.

AbstractA mapping T of a metric space $$\left( X,d\right) $$ X , d into a metric space $$ \left( Y,\rho \right) $$ Y , ρ is called restrictive Lipschitz if there exist: a positive decreasing to zero sequence $$\left( t_{n}:n\in \mathbb {N} \right) $$ t n : n ∈ N and a nonnegative sequence $$\left( L_{n}:n\in \mathbb {N}\right) ,$$ L n : n ∈ N , with $$L:=\liminf _{n\rightarrow \infty }L_{n}<\infty ,$$ L : = lim inf n → ∞ L n < ∞ , such that for all $$ x,y\in X,$$ x , y ∈ X , $$n\in \mathbb {N}$$ n ∈ N $$\begin{aligned} d\left( x,y\right) =t_{n}\Longrightarrow \rho \left( Tx,Ty\right) \le L_{n}t_{n}\text {.} \end{aligned}$$ d x , y = t n ⟹ ρ T x , T y ≤ L n t n . Using a basis property of the sequence $$\left( t_{n}:n\in \mathbb {N}\right) $$ t n : n ∈ N (Lemma 1), we prove that if T is a continuous and restrictive Lipschitz mapping of a complete metrically convex space $$\left( X,d\right) $$ X , d into a metric space $$\left( Y,\rho \right) ,$$ Y , ρ , then T is Lipschitz continuous with the constant L, that is $$\begin{aligned} \rho \left( Tx,Ty\right) \le Ld\left( x,y\right) , \quad x,y\in X, \end{aligned}$$ ρ T x , T y ≤ L d x , y , x , y ∈ X , and, in the case when the set $$\left\{ n\in \mathbb {N}:L_{n}<L\right\} $$ n ∈ N : L n < L is infinite, even essentially more, namely $$\begin{aligned} \rho \left( Tx,Ty\right) \le L\alpha \left( d\left( x,y\right) \right) , \quad x,y\in X, \end{aligned}$$ ρ T x , T y ≤ L α d x , y , x , y ∈ X , where the function $$\alpha :\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) $$ α : 0 , ∞ → 0 , ∞ is continuous, increasing, concave (so subadditive) and such that $$\alpha \left( t\right) <t$$ α t < t for all $$t>0$$ t > 0 . This result leads to the following fixed-point principle: Every continuous selfmapping T of a nonempty metrically convex complete metric space $$ \left( X,d\right) $$ X , d that is restrictive Lipschitz with a sequence $$\left( L_{n}:n\in \mathbb {N}\right) ,$$ L n : n ∈ N , such that$$\ 0\,\le L_{n}<1\ (n\in \mathbb {N) }$$ 0 ≤ L n < 1 ( n ∈ N ) and $$\liminf _{n\rightarrow \infty }L_{n}\le 1,$$ lim inf n → ∞ L n ≤ 1 , has a unique fixed point, and either it is a Banach contraction, or there is an increasing concave function $$\alpha :\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) $$ α : 0 , ∞ → 0 , ∞ , such that $$\alpha \left( t\right) <t$$ α t < t for $$t>0$$ t > 0 and $$\begin{aligned} d\left( Tx,Ty\right) \le \alpha \left( d\left( x,y\right) \right) , \quad x,y\in X. \end{aligned}$$ d T x , T y ≤ α d x , y , x , y ∈ X . Some applications of these results to the theory of iterative functional equations are proposed.

SUMMARY In recent years, there has been increasing interest in theoretical descriptions of seismicity in terms of statistical physics. Most aspects of these studies are encompassed by the concept of ‘intermittent criticality’, in which a region alternately approaches and retreats from a critical point. In the present study, we analyze a descriptor of seismic activity that acts as a measure of the criticality of a system, such that its variations can be associated with changes in the state of the system. As some classical methods of analysis are not suitable for dealing with some of the features of complex systems such as the Earth’s crust, we derive the probability distribution of the magnitude by maximizing a nonextensive generalization of the Boltzmann-Gibbs entropy given by the Tsallis entropy. In particular, the shape parameter q of this distribution, called the entropic index, characterizes the subadditive q &gt; 1 and superadditive q &lt; 1 regimes. Following the Bayesian approach for parameter estimation, we examine the seismic activity that has affected two seismogenic areas in central Italy that were hit recently by destructive earthquakes: L’Aquila in 2009, and Amatrice-Norcia in 2016. To analyze in detail the variations of the q index and the entropy, we estimate these for time windows of a fixed number of events that shift at each new event. Both the q index and the Tsallis entropy show significant and lasting decreases before the first strong earthquake in the sequences, and sudden increases after them. This indicates that these quantities can be considered as indicators of the level of concentration of energy, and hence of the activation state of the systems. More reliable results need to come from further studies of different cases in different seismotectonic settings.

Nous nous intéressons à l'étude des propriétés théoriques des équations récurrentes stochastiques (SRE) et de leurs applications en finance. Ces modèles sont couramment utilisés en économétrie, y compris en économétrie de la finance, pour styliser la dynamique d'une variété de processus tels que la volatilité des rendements financiers. Cependant, la structure de probabilité ainsi que les propriétés statistiques de ces modèles sont encore mal connues, particulièrement lorsque le modèle est considéré en dimension infinie ou lorsqu'il est généré par un processus non indépendant. Ces deux caractéristiques entraînent de formidables difficultés à l'étude théorique de ces modèles. Dans ces contextes, nous nous intéressons à l'existence de solutions stationnaires, ainsi qu'aux propriétés statistiques et probabilistes de ces solutions.Nous établissons de nouvelles propriétés sur la trajectoire de la solution stationnaire des SREs que nous exploitons dans l'étude des propriétés asymptotiques de l'estimateur du quasi-maximum de vraisemblance (QMLE) des modèles de volatilité conditionnelle de type GARCH. En particulier, nous avons étudié la stationnarité et l'inférence statistique des modèles GARCH(p,q) semi-forts dans lesquels le processus d'innovation n'est pas nécessairement indépendant. Nous établissons la consistance du QMLE des GARCH (p,q) semi-forts sans hypothèses d'existence de moment, couramment supposée pour ces modèles, sur la distribution stationnaire. De même, nous nous sommes intéressés aux modèles GARCH à deux facteurs (GARCH-MIDAS); un facteur de volatilité à long terme et un autre à court terme. Ces récents modèles introduits par Engle et al. (2013) ont la particularité d'avoir des solutions stationnaires avec des distributions à queue épaisse. Ces modèles sont maintenant fréquemment utilisés en économétrie, cependant, leurs propriétés statistiques n'ont pas reçu beaucoup d'attention jusqu'à présent. Nous montrons la consistance et la normalité asymptotique du QMLE des modèles GARCH-MIDAS et nous proposons différentes procédures de test pour évaluer la présence de volatilité à long terme dans ces modèles. Nous illustrons nos résultats avec des simulations et des applications sur des données financières réelles.Enfin, nous étendons le résultat de Kesten (1975) sur le taux de croissance des séquences additives aux processus superadditifs. Nous déduisons de ce résultat des généralisations de la propriété de contraction des matrices aléatoires aux produits d'opérateurs stochastiques. Nous utilisons ces résultats pour établir des conditions nécessaires et suffisantes d'existence de solutions stationnaires du modèle affine à coefficients positifs des SREs dans l'espace des fonctions continues. Cette classe de modèles regroupe la plupart des modèles de volatilité conditionnelle, y compris les GARCH fonctionnels
We are interested in the theoretical properties of Stochastic Recurrent Equations (SRE) and their applications in finance. These models are widely used in econometrics, including financial econometrics, to explain the dynamics of various processes such as the volatility of financial returns. However, the probability structure and statistical properties of these models are still not well understood, especially when the model is considered in infinite dimensions or driven by non-independent processes. These two features lead to significant difficulties in the theoretical study of these models. In this context, we aim to explore the existence of stationary solutions and the statistical and probabilistic properties of these solutions.We establish new properties on the trajectory of the stationary solution of SREs, which we use to study the asymptotic properties of the quasi-maximum likelihood estimator (QMLE) of GARCH-type (generalized autoregressive conditional heteroskedasticity) conditional volatility models. In particular, we study the stationarity and statistical inference of semi-strong GARCH(p,q) models where the innovation process is not necessarily independent. We establish the consistency of the QMLE of semi-strong GARCHs without assuming the commonly used condition that the stationary distribution admits a small-order moment. In addition, we are interested in the two-factor volatility GARCH models (GARCH-MIDAS); a long-run, and a short-run volatility. These models were recently introduced by Engle et al. (2013) and have the particularity to admit stationary solutions with heavy-tailed distributions. These models are now widely used but their statistical properties have not received much attention. We show the consistency and asymptotic normality of the QMLE of the GARCH-MIDAS models and provide various test procedures to evaluate the presence of long-run volatility in these models. We also illustrate our results with simulations and applications to real financial data.Finally, we extend a result of Kesten (1975) on the growth rate of additive sequences to superadditive processes. From this result, we derive generalizations of the contraction property of random matrices to products of stochastic operators. We use these results to establish necessary and sufficient conditions for the existence of stationary solutions of the affine case with positive coefficients of SREs in the space of continuous functions. This class of models includes most conditional volatility models, including functional GARCHs