scholarly journals On involution kernels and large deviations principles on $ \beta $-shifts

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Victor Vargas
Keyword(s):  
Large Deviations ◽  
Gibbs State ◽  
Natural Extension ◽  
Rate Function ◽  
Hölder Continuous ◽  
Large Deviations Principle ◽  
Ruelle Operator ◽  
The Family ◽  
In Series ◽  
Continuous Potential

<p style='text-indent:20px;'>Consider <inline-formula><tex-math id="M2">\begin{document}$ \beta &gt; 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \lfloor \beta \rfloor $\end{document}</tex-math></inline-formula> its integer part. It is widely known that any real number <inline-formula><tex-math id="M4">\begin{document}$ \alpha \in \Bigl[0, \frac{\lfloor \beta \rfloor}{\beta - 1}\Bigr] $\end{document}</tex-math></inline-formula> can be represented in base <inline-formula><tex-math id="M5">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> using a development in series of the form <inline-formula><tex-math id="M6">\begin{document}$ \alpha = \sum_{n = 1}^\infty x_n\beta^{-n} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M7">\begin{document}$ x = (x_n)_{n \geq 1} $\end{document}</tex-math></inline-formula> is a sequence taking values into the alphabet <inline-formula><tex-math id="M8">\begin{document}$ \{0,\; ...\; ,\; \lfloor \beta \rfloor\} $\end{document}</tex-math></inline-formula>. The so called <inline-formula><tex-math id="M9">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-shift, denoted by <inline-formula><tex-math id="M10">\begin{document}$ \Sigma_\beta $\end{document}</tex-math></inline-formula>, is given as the set of sequences such that all their iterates by the shift map are less than or equal to the quasi-greedy <inline-formula><tex-math id="M11">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-expansion of <inline-formula><tex-math id="M12">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>. Fixing a Hölder continuous potential <inline-formula><tex-math id="M13">\begin{document}$ A $\end{document}</tex-math></inline-formula>, we show an explicit expression for the main eigenfunction of the Ruelle operator <inline-formula><tex-math id="M14">\begin{document}$ \psi_A $\end{document}</tex-math></inline-formula>, in order to obtain a natural extension to the bilateral <inline-formula><tex-math id="M15">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-shift of its corresponding Gibbs state <inline-formula><tex-math id="M16">\begin{document}$ \mu_A $\end{document}</tex-math></inline-formula>. Our main goal here is to prove a first level large deviations principle for the family <inline-formula><tex-math id="M17">\begin{document}$ (\mu_{tA})_{t&gt;1} $\end{document}</tex-math></inline-formula> with a rate function <inline-formula><tex-math id="M18">\begin{document}$ I $\end{document}</tex-math></inline-formula> attaining its maximum value on the union of the supports of all the maximizing measures of <inline-formula><tex-math id="M19">\begin{document}$ A $\end{document}</tex-math></inline-formula>. The above is proved through a technique using the representation of <inline-formula><tex-math id="M20">\begin{document}$ \Sigma_\beta $\end{document}</tex-math></inline-formula> and its bilateral extension <inline-formula><tex-math id="M21">\begin{document}$ \widehat{\Sigma_\beta} $\end{document}</tex-math></inline-formula> in terms of the quasi-greedy <inline-formula><tex-math id="M22">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-expansion of <inline-formula><tex-math id="M23">\begin{document}$ 1 $\end{document}</tex-math></inline-formula> and the so called involution kernel associated to the potential <inline-formula><tex-math id="M24">\begin{document}$ A $\end{document}</tex-math></inline-formula>.</p>

Large deviations for the time of ruin

1999 ◽  
Vol 36 (3) ◽  
pp. 733-746 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Large Deviations ◽  
Positive Real Number ◽  
Rate Function ◽  
Positive Real ◽  
Large Deviations Principle ◽  
Borel Sets ◽  
Time Of Ruin ◽  
The Family ◽  
The Time Of Ruin ◽  
Simulation Problem

Let {Yn | n=1,2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Yn > M} (T = +∞ if Yn ≤ M for n=1,2,…). We are interested in the ruin probabilities for large M. Define the family of measures {PM | M > 0} by PM(B) = P(T/M ∊ B) for B ∊ ℬ (ℬ = Borel sets of ℝ). We prove that for a wide class of processes {Yn}, the family {PM} satisfies a large deviations principle. The rate function will correspond to the approximation P(T/M ≈ x) ≈ P(Y⌈xM⌉/M ≈ 1) for x > 0. We apply the result to a simulation problem.

Large deviations for the time of ruin

1999 ◽  
Vol 36 (03) ◽  
pp. 733-746 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Large Deviations ◽  
Positive Real Number ◽  
Rate Function ◽  
Positive Real ◽  
Large Deviations Principle ◽  
Borel Sets ◽  
Time Of Ruin ◽  
The Family ◽  
The Time Of Ruin ◽  
Simulation Problem

Let {Y n | n=1,2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Y n &gt; M} (T = +∞ if Y n ≤ M for n=1,2,…). We are interested in the ruin probabilities for large M. Define the family of measures {P M | M &gt; 0} by P M (B) = P(T/M ∊ B) for B ∊ ℬ (ℬ = Borel sets of ℝ). We prove that for a wide class of processes {Y n }, the family {P M } satisfies a large deviations principle. The rate function will correspond to the approximation P(T/M ≈ x) ≈ P(Y ⌈xM⌉/M ≈ 1) for x &gt; 0. We apply the result to a simulation problem.

scholarly journals Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets

2015 ◽  
Vol 37 (1) ◽  
pp. 79-102 ◽  
Author(s):  
THIAGO BOMFIM ◽  
PAULO VARANDAS
Keyword(s):  
Large Deviations ◽  
Gibbs Measures ◽  
Rate Function ◽  
Topological Pressure ◽  
Hölder Continuous ◽  
Continuous Potential ◽  
Hyperbolic Flows ◽  
Time Averages ◽  
Set Of Points ◽  
Level 2

In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if$f$has an expanding repeller and$\unicode[STIX]{x1D719}$is a Hölder continuous potential, we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval$I\subset \mathbb{R}$can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we deduce that most irregular sets for maps with the specification property have topological pressure strictly smaller than the whole system. Some extensions to a non-uniformly hyperbolic setting, level-2 irregular sets and hyperbolic flows are also given.

scholarly journals Run-and-Tumble Motion: The Role of Reversibility

2021 ◽  
Vol 183 (3) ◽  
Author(s):  
Bart van Ginkel ◽  
Bart van Gisbergen ◽  
Frank Redig
Keyword(s):  
Free Energy ◽  
Diffusion Coefficient ◽  
Large Deviations ◽  
Energy Function ◽  
Internal State ◽  
Rate Function ◽  
Free Energy Function ◽  
Large Deviations Principle ◽  
Preferred Direction ◽  
State Process

AbstractWe study a model of active particles that perform a simple random walk and on top of that have a preferred direction determined by an internal state which is modelled by a stationary Markov process. First we calculate the limiting diffusion coefficient. Then we show that the ‘active part’ of the diffusion coefficient is in some sense maximal for reversible state processes. Further, we obtain a large deviations principle for the active particle in terms of the large deviations rate function of the empirical process corresponding to the state process. Again we show that the rate function and free energy function are (pointwise) optimal for reversible state processes. Finally, we show that in the case with two states, the Fourier–Laplace transform of the distribution, the moment generating function and the free energy function can be computed explicitly. Along the way we provide several examples.

scholarly journals Large deviations for randomly connected neural networks: II. State-dependent interactions

2018 ◽  
Vol 50 (3) ◽  
pp. 983-1004 ◽  
Author(s):  
Tanguy Cabana ◽  
Jonathan D. Touboul
Keyword(s):  
Neural Networks ◽  
Large Deviations ◽  
Rate Function ◽  
Network Size ◽  
Kuramoto Model ◽  
Empirical Measure ◽  
Large Deviations Principle ◽  
Stochastic Version ◽  
State Dependent ◽  
Spatially Extended

Abstract We continue the analysis of large deviations for randomly connected neural networks used as models of the brain. The originality of the model relies on the fact that the directed impact of one particle onto another depends on the state of both particles, and they have random Gaussian amplitude with mean and variance scaling as the inverse of the network size. Similarly to the spatially extended case (see Cabana and Touboul (2018)), we show that under sufficient regularity assumptions, the empirical measure satisfies a large deviations principle with a good rate function achieving its minimum at a unique probability measure, implying, in particular, its convergence in both averaged and quenched cases, as well as a propagation of a chaos property (in the averaged case only). The class of model we consider notably includes a stochastic version of the Kuramoto model with random connections.

scholarly journals Large Deviations for the Single-Server Queue and the Reneging Paradox

2021 ◽  
Author(s):  
Rami Atar ◽  
Amarjit Budhiraja ◽  
Paul Dupuis ◽  
Ruoyu Wu
Keyword(s):  
Large Deviations ◽  
Decay Rate ◽  
Sample Path ◽  
Arrival Rate ◽  
Rate Function ◽  
Service Rate ◽  
Single Server ◽  
Large Deviations Principle ◽  
Long Run ◽  
The Individual

For the M/M/1+M model at the law-of-large-numbers scale, the long-run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: the decay rate of the probability that the long-run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.

scholarly journals Large deviations for randomly connected neural networks: I. Spatially extended systems

2018 ◽  
Vol 50 (3) ◽  
pp. 944-982 ◽  
Author(s):  
Tanguy Cabana ◽  
Jonathan D. Touboul
Keyword(s):  
Neural Networks ◽  
Large Deviations ◽  
Rate Function ◽  
Network Size ◽  
Empirical Measure ◽  
Interaction Coefficients ◽  
Large Deviations Principle ◽  
Implicit Equation ◽  
Random Interaction ◽  
Spatially Extended

Abstract In a series of two papers, we investigate the large deviations and asymptotic behavior of stochastic models of brain neural networks with random interaction coefficients. In this first paper, we take into account the spatial structure of the brain and consider first the presence of interaction delays that depend on the distance between cells and then the Gaussian random interaction amplitude with a mean and variance that depend on the position of the neurons and scale as the inverse of the network size. We show that the empirical measure satisfies a large deviations principle with a good rate function reaching its minimum at a unique spatially extended probability measure. This result implies an averaged convergence of the empirical measure and a propagation of chaos. The limit is characterized through a complex non-Markovian implicit equation in which the network interaction term is replaced by a nonlocal Gaussian process with a mean and covariance that depend on the statistics of the solution over the whole neural field.

scholarly journals Large deviations for the empirical measure of heavy-tailed Markov renewal processes

2016 ◽  
Vol 48 (3) ◽  
pp. 648-671 ◽  
Author(s):  
Mauro Mariani ◽  
Lorenzo Zambotti
Keyword(s):  
Large Deviations ◽  
Finite Graph ◽  
Rate Function ◽  
Empirical Measure ◽  
Markov Renewal Process ◽  
Large Deviations Principle ◽  
Arrival Times ◽  
Heavy Tailed Distributions ◽  
Markov Renewal Processes ◽  
Heavy Tailed

Abstract A large deviations principle is established for the joint law of the empirical measure and the flow measure of a Markov renewal process on a finite graph. We do not assume any bound on the arrival times, allowing heavy-tailed distributions. In particular, the rate function is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behaviour highly different from what one may guess with a heuristic Donsker‒Varadhan analysis of the problem.

scholarly journals Multipopulation Spin Models: A View from Large Deviations Theoretic Window

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Alex Akwasi Opoku ◽  
Godwin Osabutey
Keyword(s):  
Large Deviations ◽  
Probability Distributions ◽  
Mean Field ◽  
Variational Formula ◽  
Rate Function ◽  
Large Deviations Principle ◽  
Spin Models ◽  
Empirical Measures ◽  
Rate Functions ◽  
Empirical Means

This paper studies large deviations properties of vectors of empirical means and measures generated as follows. Consider a sequence X1,X2,…,Xn of independent and identically distributed random variables partitioned into d-subgroups with sizes n1,…,nd. Further, consider a d-dimensional vector mn whose coordinates are made up of the empirical means of the subgroups. We prove the following. (1) The sequence of vector of empirical means mn satisfies large deviations principle with rate n and rate function I, when the sequence X1,X2,…,Xn is Rl valued, with l≥1. (2) Similar large deviations results hold for the corresponding sequence of vector of empirical measures Ln if Xi’s, i=1,2,…,n, take on finitely many values. (3) The rate functions for the above large deviations principles are convex combinations of the corresponding rate functions arising from the large deviations principles of the coordinates of mn and Ln. The probability distributions used in the convex combinations are given by α=(α1,…,αd)=limn→∞1/n(n1,…,nd). These results are consequently used to derive variational formula for the thermodynamic limit for the pressure of multipopulation Curie-Weiss (I. Gallo and P. Contucci (2008), and I. Gallo (2009)) and mean-field Pott’s models, via a version of Varadhan’s integral lemma for an equicontinuous family of functions. These multipopulation models serve as a paradigm for decision-making context where social interaction and other socioeconomic attributes of individuals play a crucial role.

Continuous-time Markov additive processes: Composition of large deviations principles and comparison between exponential rates of convergence

2001 ◽  
Vol 38 (4) ◽  
pp. 917-931 ◽  
Author(s):  
Claudio Macci
Keyword(s):  
Large Deviations ◽  
Continuous Time ◽  
Fluid Model ◽  
Variational Formula ◽  
Rates Of Convergence ◽  
Rate Function ◽  
Large Deviations Principle ◽  
Additive Process ◽  
Additive Processes ◽  
Markov Additive Processes

We consider a continuous-time Markov additive process (Jt,St) with (Jt) an irreducible Markov chain on E = {1,…,s}; it is known that (St/t) satisfies the large deviations principle as t → ∞. In this paper we present a variational formula H for the rate function κ∗ and, in some sense, we have a composition of two large deviations principles. Moreover, under suitable hypotheses, we can consider two other continuous-time Markov additive processes derived from (Jt,St): the averaged parameters model (Jt,St(A)) and the fluid model (Jt,St(F)). Then some results of convergence are presented and the variational formula H can be employed to show that, in some sense, the convergences for (Jt,St(A)) and (Jt,St(F)) are faster than the corresponding convergences for (Jt,St).

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