Bernoulli Shift Research Articles

Dependence properties of stochastic volatility models

The concepts of physical dependence and approximability have been extensively used over the past two decades to quantify nonlinear dependence in time series. We show that most stochastic volatility models satisfy both dependence conditions, even if their realizations take values in abstract Hilbert spaces, thus covering univariate, multi‐variate and functional models. Our results can be used to apply to general stochastic volatility models a multitude of inferential procedures established for Bernoulli shifts.

Dowker’s ergodic theorem by the Chacon–Ornstein theorem

Dowker’s ergodic theorem by the Chacon–Ornstein theorem

Approach to hyperuniformity of steady states of facilitated exclusion processes

We consider the fluctuations in the number of particles in a box of size Ld in Zd , d⩾1 , in the (infinite volume) translation invariant stationary states of the facilitated exclusion process, also called the conserved lattice gas model. When started in a Bernoulli (product) measure at density ρ, these systems approach, as t→∞ , a ‘frozen’ state for ρ⩽ρc , with ρc=1/2 for d = 1 and ρc<1/2 for d⩾2 . At ρ=ρc the limiting state is, as observed by Hexner and Levine, hyperuniform, that is, the variance of the number of particles in the box grows slower than Ld . We give a general description of how the variances at different scales of L behave as ρ↗ρc . On the largest scale, L≫L2 , the fluctuations are normal (in fact the same as in the original product measure), while in a region L1≪L≪L2 , with both L 1 and L 2 going to infinity as ρ↗ρc , the variance grows faster than normal. For 1≪L≪L1 the variance is the same as in the hyperuniform system. (All results discussed are rigorous for d = 1 and based on simulations for d⩾2 .)

Absolutely continuous invariant measures for random dynamical systems of beta-transformations

We consider an independent and identically distributed (i.i.d.) random dynamical system of simple linear transformations on the unit interval Tβ(x)=βx (mod 1), x∈[0,1] , β > 0, which are the so-called beta-transformations. For such a random dynamical system, including the case that it is generated by uncountably many maps, we give an explicit formula for the density function of a unique stationary measure under the assumption that the random dynamics is expanding in mean. As an application, in the case that the random dynamics is generated by finitely many maps and the maps are chosen according to a Bernoulli measure, we show that the density function is analytic as a function of parameter in the Bernoulli measure and give its derivative explicitly. Furthermore, for a non-i.i.d. random dynamical system of beta-transformations, we also give an explicit formula for the random densities of a unique absolutely continuous invariant measure under a certain strong expanding condition or under the assumption that the maps randomly chosen are close to the beta-transformation for a non-simple number in the sense of parameter β.

On Λ-Fractional fluid mechanics

Λ-fractional analysis has already been presented as the only fractional analysis conforming with the Differential Topology prerequisites. That is, the Leibniz rule and chain rule do not apply to other fractional derivatives; This, according to Differential Topology, makes the definition of a differential impossible for these derivatives. Therefore, this leaves Λ-fractional analysis the only analysis generating differential geometry necessary to establish the governing laws in physics and mechanics. Hence, it is most necessary to use Λ-fractional derivative and Λ-fractional transformation to describe fractional mathematical models. Other fractional “derivatives” are not proper derivatives, according to Differential Topology; they are just operators. This fact makes their application to mathematical problems questionable while Λ-derivative faces no such problems. Basic Fluid Mechanics equations are studied and revised under the prism of Λ-Fractional Continuum Mechanics (Λ-FCM). Extending the already presented principles of Continuum Mechanics in the area of solids into the area of fluids, the basic Λ-fractional fluid equations concerning the Navier-Stokes, Euler, and Bernoulli flows are derived, and the Λ-fractional Darcy’s flow in porous media is studied. Since global minimization of the various fields is accepted only in the Λ-fractional analysis, shocks in the Λ-fractional motion of fluids are exhibited.

Asymptotic normality of spectral means of Hilbert space valued random processes

A variety of statistics for functional time series allows for a representation as weighted average of corresponding periodogram operators over the frequency domain. We study consistency and asymptotic normality of such spectral mean estimators under mild assumptions. We show that weak convergence of these estimators can be deduced from the (joint) weak convergence of the sample autocovariance operators. The latter is established for a large class of weakly dependent functional time series, which admit expansions as Bernoulli shifts and the weak dependence is quantified by the condition of L4-m-approximability.

Chaotic Phenomena for Generalised N-centre Problems

We study a class of singular dynamical systems which generalise the classical N-centre problem of Celestial Mechanics to the case in which the configuration space is a Riemannian surface. We investigate the existence of topological conjugation with the archetypal chaotic dynamical system, the Bernoulli shift. After providing infinitely many geometrically distinct and collision-less periodic solutions, we encode them in bi-infinite sequences of symbols. Solutions are obtained as minimisers of the Maupertuis functional in suitable free homotopy classes of the punctured surface, without any collision regularisation. For any sufficiently large value of the energy, we prove that the generalised N-centre problem admits a symbolic dynamics. Moreover, when the Jacobi-Maupertuis metric curvature is negative, we construct chaotic invariant subsets.

On the failure of Ornstein theory in the finitary category

We show the invalidity of finitary counterparts for three classification theorems: The preservation of being a Bernoulli shift through factors, Sinai’s factor theorem, and the weak Pinsker property. We construct a finitary factor of an i.i.d. process which is not finitarily isomorphic to an i.i.d. process, showing that being finitarily Bernoulli is not preserved through finitary factors. This refutes a conjecture of M. Smorodinsky [Finitary isomorphism of m m -dependent processes, Amer. Math. Soc., Birkhauser, Providence, RI, 1992, pp. 373–376], which was first suggested by D. Rudolph [A characterization of those processes finitarily isomorphic to a Bernoulli shift, Birkhäuser, Boston, Mass., 1981, pp. 1–64]. We further show that any ergodic system is isomorphic to a process none of whose finitary factors are i.i.d. processes, and in particular, there is no general finitary Sinai’s factor theorem for ergodic processes. Another consequence of this result is the invalidity of a finitary weak Pinsker property, answering a question of G. Pete and T. Austin [Math. Inst. Hautes Études Sci. 128 (2018), 1–119].

On discrete properties of Bernoulli shift

Monounary algebras are the most simple type of an algebraic structure. Oriented graphs with one outgoing arrow from every vertex represent them. The aim of this paper is to point out the interdisciplinary relationships concerning this structure. Bernoulli shift is a paradigmatic mapping in dynamical systems. It is also called dyadic, bit shift, doubling or sawtooth. We offer a look at the properties of this mapping via monounary algebras.

Proving chaos for a system of coupled logistic maps: A topological approach.

In the work, we prove the presence of chaotic dynamics, for suitable values of the model parameters, for the discrete-time system, composed of two coupled logistic maps, as formulated in Yousefi et al. [Discrete Dyn. Nat. Soc. 5, 161-177 (2000)], which describes two interdependent economies, characterized by two competitive markets each, with a demand link between them. In particular, we rely on the SAP (Stretching Along the Paths) method, based on a stretching relation for maps defined on sets homeomorphic to the unit square and expanding the paths along one direction. Such technique is topological in nature and allows to establish the existence of a semiconjugacy between the considered dynamical system and the Bernoulli shift, so that the main features related to the chaos of the latter (e.g., the positivity of the topological entropy) are transmitted to the former. In more detail, we apply the SAP method both to the first and to the second iterate of the map associated with our system, and we show how dealing with the second iterate, although being more demanding in terms of computations, allows for a larger freedom in the sign and in the variation range of the linking parameters for which chaos emerges. Moreover, the latter choice guarantees a good agreement with the numerical simulations, which highlight the presence of a chaotic attractor under the conditions derived for the applicability of the SAP technique to the second iterate of the map.

First ℓ2-Betti numbers and proper proximality

We show that for a countable exact group, having positive first ℓ2-Betti number implies proper proximality in this sense of [3]. This is achieved by showing a cocycle superrigidity result for Bernoulli shifts of non-properly proximal groups. We also obtain that Bernoulli shifts of countable, nonamenable, i.c.c., exact, non-properly proximal groups are OE-superrigid.

Sinai factors of nonsingular systems: Bernoulli shifts and Anosov flows

Sinai factors of nonsingular systems: Bernoulli shifts and Anosov flows

Doubling properties of self-similar measures and Bernoulli measures on self-affine Sierpinski sponges

Doubling properties of self-similar measures and Bernoulli measures on self-affine Sierpinski sponges

Exploring a family of Bernoulli-like shift chaotic maps and its amplitude control

Chaos theory is a branch of mathematics that studies systems highly sensitive to initial conditions. Various methods for analyzing and controlling these systems, such as bifurcation theory and amplitude control, have been developed and applied in diverse fields, including cryptography, physics, and engineering.In this study, we introduce a new class of Bernoulli-Like Shift (BLS) maps. These maps are generated by using a specific class of Piecewise Linear (PWL) maps combined with a modulo operation. We conduct a fixed point analysis by establishing propositions and corollaries, which outline the conditions that guarantee the presence or absence of fixed points. Additionally, we provide numerical examples to validate these theoretical results.Utilizing bifurcation diagrams and Lyapunov exponent calculations, we demonstrate that the range of chaotic behavior in the BLS maps is broader than those of the logistic and Bernoulli Shift maps. Moreover, we show that the BLS maps possess higher values of Lyapunov exponents and a better distribution than both the Bernoulli shift and logistic maps.Lastly, we introduce a transformation that generates a modified BLS map, allowing for amplitude control. This modification enables the regulation of amplitude while maintaining constant Lyapunov exponents.

Factor-of-iid balanced orientation of non-amenable graphs

We show that if a non-amenable, quasi-transitive, unimodular graph G has all degrees even then it has a factor-of-iid balanced orientation, meaning each vertex has equal in- and outdegree. This result involves extending earlier spectral-theoretic results on Bernoulli shifts to the Bernoulli graphings of quasi-transitive, unimodular graphs.As a consequence, we also obtain that when G is regular (of either odd or even degree) and bipartite, it has a factor-of-iid perfect matching. This generalizes a result of Lyons and Nazarov beyond transitive graphs.

Assessment of ship position probability in the geometrical figure of position lines errors

Accurate and reliable determination of the ship position coordinates is one of the most important tasks for ensuring the safety of navigation at sea. A significant condition that contributes to the effective solution of this task is the use of redundancy in measurements of navigation parameters. The implementation of this redundancy allows the navigator or automatic navigation aid to control the trueness of measurements; eliminate gross and systematic errors; reduce the effect of random errors; apply various methods of analysis and algorithms for processing information in order to obtain the most accurate and reliable ship position. Adequate redundancy of navigation measurements makes it possible to obtain asymptotically unbiased estimates of coordinates, as well as to solve other navigation tasks based on redundant data. The results of a study on the probability of finding the true coordinates of the ship’s position in a geometric figure formed by random measurement errors are presented. It has been shown in the study that a single redundancy of measurements is not enough to achieve the required level of probability. Elements of classical combinatorics based on the Bernoulli scheme and including Newton’s binomial and Pascal’s triangle as tools are used as a mathematical apparatus for the study. These tools allow us to effectively analyze combinations of mathematical signs, such as plus and minus, of random measurement errors, reducing them into binary sequences, which are described by binomial coefficients. It has been argued in the paper that the low level of probability of finding the true values of the coordinates in the geometrical figure of measurement errors is a strong signal that any method of averaging the measurement results can lead to asymptotically biased estimates of the ship position coordinates. This is analogous to the influence of systematic errors on the coordinates within a certain time interval. From the point of view of navigational safety, a low probability can be qualified as a near-miss situation, which can lead to incorrect control of the ship position. The probabilistic criteria for the necessary redundancy of measurements to approach asymptotically unbiased estimates of ship position are presented in the paper.

Distribution of δ-connected components of self-affine sponges of Lalley-Gatzouras type

Let (E,ρ) be a metric space and let hE(δ) be the cardinality of the set of δ-connected components of E. In literature, in case of that E is a self-conformal set satisfying the open set condition or E is a self-affine Sierpiński sponge, necessary and sufficient condition is given for the validity of the relationhE(δ)≍δ−dimB⁡E, when δ→0. In this paper, we generalize the above result to self-affine sponges of Lalley-Gatzouras type; actually in this case, we show that there exists a Bernoulli measure μ such that for any cylinder W, it holds thathW(δ)≍μ(W)δ−dimB⁡E, when δ→0.

EQUILIBRIUM STATES FOR CENTER ISOMETRIES

AbstractWe develop a geometric method to establish the existence and uniqueness of equilibrium states associated to some Hölder potentials for center isometries (as are regular elements of Anosov actions), in particular, the entropy maximizing measure and the SRB measure. A characterization of equilibrium states in terms of their disintegrations along stable and unstable foliations is also given. Finally, we show that the resulting system is isomorphic to a Bernoulli scheme.

Single-Server Queuing-Inventory Systems with Negative Customers and Catastrophes in the Warehouse

In this paper, we studied single-server models of queuing-inventory systems (QIS) with catastrophes in the warehouse part and negative customers (n-customers) in service facility. Consumer customers (c-customers) that arrived to buy inventory can be queued in an infinite buffer. Under catastrophes, all inventory of the system is destroyed but customers in the system (on server or in buffer) are still waiting for replenishment of stocks. Upon arrival of n-customer one c-customer is pushed out, if any. One of two replenishment policies (RP) can be used in the system: either (s,S) or randomized. In the investigated QISs, a hybrid service scheme was used: if upon arrival of the c-customer, the inventory level is zero, then according to the Bernoulli scheme, this customer is either lost (lost sale scheme) or joining the queue (backorder scheme). Mathematical models of the investigated QISs were constructed as two-dimensional Markov chains (2D MC). Ergodicity conditions of the investigated QISs were obtained, and the matrix-analytic method (MAM) was used to calculate the steady-state probabilities of the constructed 2D MCs. Formulas for performance measures were found and the results of numerical experiments are presented.

Validation of Reliability Indices during Experimental Development of a Complex Technical Series System

Introduction. The article studies the problem of validating the specified levels of reliability during experimental development of a complex technical series system. Such tasks arise when it is required to make a decision on testing the system as part of a larger one or on the completion of experimental development and the start of series production. The study is aimed at validating the reduction of the experimental development time. The task is to determine whether the hypothesis Но is accepted or rejected.Materials and Methods. To implement the research objective and task, a critical area described by the inequality was constructed based on the test results. The formulation of the requirements validation task was based on well-known approaches to testing statistical hypotheses. The conceptual apparatus of information theory, probability, and statistics was involved. The theoretical and applied literature on mathematical methods in reliability theory was studied. The particular tasks of the work were solved by known ways. Thus, the probability of obtaining the exact number of successful outcomes in a certain number of experiments was determined by the Bernoulli scheme. The exact confidence interval based on the binomial distribution was derived from the Clopper-Pearson relation. The theorem of A.D. Solovyov and R. A. Mirny made it possible to assess the system reliability based on the test results of its components.Results. Control rules adequate to the stage of experimental development (with insufficient data on the technical system) and the stage of series production were mathematically defined. The probability of a successful outcome when testing technical systems was represented by:– the probability of event for a system element;– confidence value;– required scope of tests.In these terms, the null and alternative hypotheses and the corresponding reliability control procedures were investigated. Two provisions were considered. The first one provided using the null confidence hypothesis Но = {Р ≥ РТ} and an alternative Н = { Р &lt; РТ} to confirm the requirements (РТ, γ) for the reliability indicator of one parameter for any (РТ, γ). In this case, one trouble-free test was enough. The second provision considered a sequential technical system with independent elements that were tested separately from the system according to the Bernoulli scheme for one parameter. We considered the requirements for the system in the form of a set of values (РТ, γ) and the requirements for any of its elements (РТi, γ). They coincided when the planned outcome of the tests corresponded to the cases when the ratio Р= lim 1≤i≤N : Р i = Р m was fulfilled, and the null alternative hypothesis was selected from the theory of statistical hypothesis testing.Discussion and Conclusions. The experimental development strategy should be implemented in two stages: the search and validation of the reliability of the elements through a series of fail-safe tests. In this case, the planned scope of tests of each element is determined taking into account the confidence probability, the lower limit of the confidence interval, and the requirements for reliability indices of one parameter of the technical system. If the use of the null confidence hypothesis is acceptable, one fail-safe test is sufficient to confirm the requirements for the reliability index.