Ergodic Sequences Research Articles

On the strong stability of ergodic iterations

Abstract We revisit processes generated by iterated random functions driven by a stationary and ergodic sequence. Such a process is called strongly stable if a random initialization exists for which the process is stationary and ergodic, and for any other initialization the difference of the two processes converges to zero almost surely. Under some mild conditions on the corresponding recursive map, without any condition on the driving sequence we show the strong stability of iterations. Several applications are surveyed such as generalized autoregression and queuing. Furthermore, new results are deduced for Langevin-type iterations with dependent noise and for multitype branching processes.

Central limit theorem under the Dedecker–Rio condition in some Banach spaces

We extend the central limit theorem under the Dedecker–Rio condition to adapted stationary and ergodic sequences of random variables taking values in a class of smooth Banach spaces. This result applies to the case of random variables taking values in Lp(μ), with 2⩽p<∞ and μ a σ-finite real measure. As an application we give a sufficient condition for empirical processes indexed by Sobolev balls to satisfy the central limit theorem, and discuss about the optimality of these conditions.

Some notes on ergodic theorem for [formula omitted]-statistics of order [formula omitted] for stationary and not necessarily ergodic sequences

In this note, we give sufficient conditions for the almost sure and the convergence in Lp of a U-statistic of order m built on a strictly stationary but not necessarily ergodic sequence.

Bounded Real Lemma for the Anisotropic Norm of Time-invariant Systems with Multiplicative Noises

We consider a discrete-time-invariant system with multiplicative noise with implementation in the state space. The exogenous disturbance is chosen from the class of time-invariant ergodic sequences of nonzero colorness. We consider the level of mean anisotropy of the exogenous disturbance to be bounded by a known value. Conditions for the anisotropic norm to be bounded by a given number are obtained in terms of solving a matrix system of inequalities with a convex constraint of a special type. It is demonstrated how, on the basis of the obtained conditions, to construct a static state control that ensures the minimum value of the anisotropic norm of the system enclosed by this control.

Joint ergodicity of sequences

A collection of integer sequences is jointly ergodic if for every ergodic measure preserving system the multiple ergodic averages, with iterates given by this collection of sequences, converge in the mean to the product of the integrals. We give necessary and sufficient conditions for joint ergodicity that are flexible enough to recover the known examples of jointly ergodic sequences and also allow us to answer some related open problems. An interesting feature of our arguments is that they avoid deep tools from ergodic theory that were previously used to establish similar results. Our approach is primarily based on an ergodic variant of a technique pioneered by Peluse and Prendiville in order to give quantitative variants for the finitary version of the polynomial Szemerédi theorem.

A nested primal–dual FISTA-like scheme for composite convex optimization problems

We propose a nested primal–dual algorithm with extrapolation on the primal variable suited for minimizing the sum of two convex functions, one of which is continuously differentiable. The proposed algorithm can be interpreted as an inexact inertial forward–backward algorithm equipped with a prefixed number of inner primal–dual iterations for the proximal evaluation and a “warm–start” strategy for starting the inner loop, and generalizes several nested primal–dual algorithms already available in the literature. By appropriately choosing the inertial parameters, we prove the convergence of the iterates to a saddle point of the problem, and provide an O(1/n) convergence rate on the primal–dual gap evaluated at the corresponding ergodic sequences. Numerical experiments on some image restoration problems show that the combination of the “warm–start” strategy with an appropriate choice of the inertial parameters is strictly required in order to guarantee the convergence to the real minimum point of the objective function.

An Ergodic Theorem for Quantum Processes with Applications to Matrix Product States

Any discrete quantum process is represented by a sequence of quantum channels. We consider ergodic quantum processes obtained by a map that takes the points along the trajectory of a discrete ergodic dynamical system to the space of quantum channels. Under a natural irreducibility condition, we obtain a theorem showing that the state under such a process converges exponentially fast to an ergodic sequence depending on the process, but independent of the initial state. As an application, we describe the thermodynamic limit of ergodic matrix product states and prove that the 2-point correlations of local observables in such states decay exponentially with their distance in the bulk.

On the Convergence of Stochastic Primal-Dual Hybrid Gradient

In this paper, we analyze the recently proposed stochastic primal-dual hybrid gradient (SPDHG) algorithm and provide new theoretical results. In particular, we prove almost sure convergence of the iterates to a solution and linear convergence with standard step sizes, independent of strong convexity constants. Our assumption for linear convergence is metric subregularity, which is satisfied for smooth and strongly convex problems in addition to many nonsmooth and/or nonstrongly convex problems, such as linear programs, Lasso, and support vector machines. In the general convex case, we prove optimal sublinear rates for the ergodic sequence and for randomly selected iterate, without bounded domain assumptions. We also provide numerical evidence showing that SPDHG with standard step sizes shows favorable and robust practical performance against its specialized strongly convex variant SPDHG-$\mu$ and other state-of-the-art algorithms including variance reduction methods and stochastic dual coordinate ascent.

Analysis of multiple data sequences with different distributions: defining common principal component axes by ergodic sequence generation and multiple reweighting composition

Principal component analysis (PCA) defines a reduced space described by PC axes for a given multidimensional-data sequence to capture the variations of the data. In practice, we need multiple data sequences that accurately obey individual probability distributions and for a fair comparison of the sequences we need PC axes that are common for the multiple sequences but properly capture these multiple distributions. For these requirements, we present individual ergodic samplings for these sequences and provide special reweighting for recovering the target distributions.

Measure invariance of ergodic symbolic systems for low-delay detection of anomalous events

This paper investigates the spectral properties of ergodic measure of symbolic systems for applications to signal processing, pattern recognition, and anomaly detection in uncertain dynamical systems. The underlying algorithm is built upon the concept of ergodic sequences of measure-preserving transformations (MPT) on probability spaces, where non-stationary probabilistic finite state automata (PFSA) are constructed using short-length windows of time-series measurements. The resulting PFSA are non-homogeneous Markov, in which spectral properties of the MPT depend on several parameters that include the window length. The paper also develops an MPT-based metric to quantify the divergence of the evolving PFSA from that of the nominal PFSA; this information is then used for pattern classification and anomaly detection with low-delay tolerance. The MPT-based methodology has been validated with experimental data generated from a laboratory apparatus that deals with detection of thermoacoustic instabilities in combustion processes, which are known to have chaotic characteristics on a time scale of milliseconds. In this application, the concepts of MPT and ergodicity have been used to develop a novel symbolic time series analysis (STSA)-based detection method, whose performance is validated by comparison with two well-known techniques, namely, hidden Markov model (HMM) and cumulative sum (CUSUM), on the same experimental data. The results consistently show superior performance of the proposed MPT-based STSA.

A splitting subgradient algorithm for solving equilibrium problems involving the sum of two bifunctions and application to cournot-nash model

In this paper we propose a splitting subgradient algorithm for solving equilibrium problems involving the sum of two bifunctions. At each iteration of the algorithm, two strongly convex subprograms are required to solve separately, one for each component bifunction. In contrast to the splitting algorithms previously proposed in Anh and Hai (Numer. Algorithms 76 (2017) 67–91) and Hai and Vinh (Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 111 (2017) 1051–1069), our algorithm is convergent for paramonotone and strongly pseudomonotone bifunctions without any Lipschitz type as well as Hölder continuity condition of the bifunctions involved. Furthermore, we show that the ergodic sequence defined by the algorithm iterates converges to a solution without paramonotonicity property. Some numerical experiments on differentiated Cournot-Nash models are presented to show the behavior of our proposed algorithm with and without ergodic.

Enhanced Lagrange Decomposition for multi-objective scalable TE in SDN

The paradigm of Software Defined Networking opens attractive perspectives for network operators in terms of traffic engineering (TE) mechanisms. Thanks to programmability of a logically centralized network controller, static optimization techniques became applicable to optimization of resource utilization. Such an approach, combined with accurate traffic prediction, enabled deployment of globally scoped, efficient solutions. However, such an approach might pose some scalability issues which must be carefully considered. The challenge is to propose a solution able to solve multi-objective problems with non-linear constraints in large-scale networks. Networks, that are comprised of numerous nodes handling millions of flows of dynamic nature.Our work is aimed at addressing these issues. A novel, energy-aware, multi-objective, mixed integer linear programming problem is formulated, and solved using the Lagrange Decomposition method. The method is enhanced by novel adoption of ergodic sequences in order to improve quality of a primal solution recovered from a dual one. The proposed heuristics is carefully assessed regarding its time-efficiency, applicability to multi-objective optimization problems and problems with constraints of non-linear nature, such as energy consumption. The presented method is of significant practical value, and as such is a direct response to traffic engineering scalability challenges in Software Defined Networks.

A spectral approach for quenched limit theorems for random hyperbolic dynamical systems

We extend the recent spectral approach for quenched limit theorems developed for piecewise expanding dynamics under general random driving to quenched random piecewise hyperbolic dynamics. For general ergodic sequences of maps in a neighborhood of a hyperbolic map we prove a quenched large deviations principle (LDP), central limit theorem (CLT), and local central limit theorem (LCLT).

A generalization of Kátai's orthogonality criterion with applications

We study properties of arithmetic sets coming from multiplicative number theory and obtain applications in the theory of uniform distribution and ergodic theory. Our main theorem is a generalization of K\'atai's orthogonality criterion. Here is a special case of this theorem: Let $a\colon\mathbb{N}\to\mathbb{C}$ be a bounded sequence satisfying $$ \sum_{n\leq x} a(pn)\overline{a(qn)} = {\rm o}(x),~\text{for all distinct primes $p$ and $q$.} $$ Then for any multiplicative function $f$ and any $z\in\mathbb{C}$ the indicator function of the level set $E=\{n\in\mathbb{N}:f(n)=z\}$ satisfies $$ \sum_{n\leq x} \mathbb{1}_E(n)a(n)={\rm o}(x). $$ With the help of this theorem one can show that if $E=\{n_1<n_2<\ldots\}$ is a level set of a multiplicative function having positive upper density, then for a large class of sufficiently smooth functions $h\colon(0,\infty)\to\mathbb{R}$ the sequence $(h(n_j))_{j\in\mathbb{N}}$ is uniformly distributed $\bmod~1$. This class of functions $h(t)$ includes: all polynomials $p(t)=a_kt^k+\ldots+a_1t+a_0$ such that at least one of the coefficients $a_1,a_2,\ldots,a_k$ is irrational, $t^c$ for any $c>0$ with $c\notin \mathbb{N}$, $\log^r(t)$ for any $r>2$, $\log(\Gamma(t))$, $t\log(t)$, and $\frac{t}{\log t}$. The uniform distribution results, in turn, allow us to obtain new examples of ergodic sequences, i.e. sequences along which the ergodic theorem holds.

Inner radius of nodal domains of quantum ergodic eigenfunctions

In this short note we show that the lower bounds of Mangoubi on the inner radius of nodal domains can be improved for quantum ergodic sequences of eigenfunctions, according to a certain power of the radius of shrinking balls on which the eigenfunctions equidistribute. We prove such improvements using a quick application of our recent results (arXiv:1606.02057), which give modified growth estimates for eigenfunctions that equidistribute on small balls. Since by the results of Han and Hezari-Riviere small scale QE holds for negatively curved manifolds on logarithmically shrinking balls, we get logarithmic improvements on the inner radius of such manifolds. We also get improvements for manifolds with ergodic geodesic flows. In addition using the small scale equidistribution results of Lester-Rudnick, one gets polynomial betterments of for toral eigenfunctions in dimensions $n \geq 3$. The results work only for a full density subsequence of eigenfunctions.

Bohr Sets in Triple Products of Large Sets in Amenable Groups

We answer a question of Hegyvari and Ruzsa concerning effective estimates of the Bohr-regularity of certain triple sums of sets with positive upper Banach densities in the integers. Our proof also works for any discrete amenable group, and it does not require all addends in the triple products we consider to have positive (left) upper Banach densities; one of the addends is allowed to only have positive upper asymptotic density with respect to a (possibly very sparse) ergodic sequence.

Möbius disjointness along ergodic sequences for uniquely ergodic actions

We show that there is an irrational rotation $Tx=x+\unicode[STIX]{x1D6FC}$ on the circle $\mathbb{T}$ and a continuous $\unicode[STIX]{x1D711}:\mathbb{T}\rightarrow \mathbb{R}$ such that for each (continuous) uniquely ergodic flow ${\mathcal{S}}=(S_{t})_{t\in \mathbb{R}}$ acting on a compact metric space $Y$, the automorphism $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$ acting on $(X\times Y,\unicode[STIX]{x1D707}\otimes \unicode[STIX]{x1D708})$ by the formula $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}(x,y)=(Tx,S_{\unicode[STIX]{x1D711}(x)}(y))$, where $\unicode[STIX]{x1D707}$ stands for the Lebesgue measure on $\mathbb{T}$ and $\unicode[STIX]{x1D708}$ denotes the unique ${\mathcal{S}}$-invariant measure, has the property of asymptotically orthogonal powers. This gives a class of relatively weakly mixing extensions of irrational rotations for which Sarnak’s conjecture on the Möbius disjointness holds for all uniquely ergodic models of $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$. Moreover, we obtain a class of ‘random’ ergodic sequences $(c_{n})\subset \mathbb{Z}$ such that if $\boldsymbol{\unicode[STIX]{x1D707}}$ denotes the Möbius function, then $$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n\leq N}g(S_{c_{n}}y)\boldsymbol{\unicode[STIX]{x1D707}}(n)=0\end{eqnarray}$$ for all (continuous) uniquely ergodic flows ${\mathcal{S}}$, all $g\in C(Y)$ and $y\in Y$.

Quantum Ergodic Sequences and Equilibrium Measures

We generalize the definition of a “quantum ergodic sequence” of sections of ample line bundles $$L \rightarrow M$$ from the case of positively curved Hermitian metrics h on L to general smooth metrics. A choice of smooth Hermitian metric h on L and a Bernstein–Markov measure $$\nu $$ on M induces an inner product on $$H^0(M, L^N)$$ . When $$||s_N||_{L^2} =1$$ , quantum ergodicity is the condition that $$|s_N(z)|^2 d\nu \rightarrow d\mu _{\varphi _{eq}} $$ weakly, where $$d\mu _{\varphi _{eq}} $$ is the equilibrium measure associated with $$(h, \nu )$$ . The main results are that normalized logarithms $$\frac{1}{N} \log |s_N|^2$$ of quantum ergodic sections tend to the equilibrium potential, and that random orthonormal bases of $$H^0(M, L^N)$$ are quantum ergodic.

Entire functions of exponential type represented by pseudo-random and random Taylor series

We study the influence of the multipliers ξ(n) on the angular distribution of zeroes of the Taylor series $${F_\xi }\left( z \right) = \sum\limits_{n \geqslant 0} {\xi \left( n \right)} \frac{{{z^n}}}{{n!}}$$ . We show that the distribution of zeroes of Fξ is governed by certain autocorrelations of the sequence ξ. Using this guiding principle, we consider several examples of random and pseudo-random sequences ξ and, in particular, answer some questions posed by Chen and Littlewood in 1967. As a by-product, we show that if ξ is a stationary random integer-valued sequence, then either it is periodic, or its spectral measure has no gaps in its support. The same conclusion is true if ξ is a complex-valued stationary ergodic sequence that takes values in a uniformly discrete set.

Asymptotic behaviour of proportions of observations in random regions determined by central order statistics from stationary processes

ABSTRACTIn this paper, we show that proportions of observations that fall into a random region determined by a given Borel set and a central order statistic converge almost surely, provided that the corresponding population quantile is unique. We also describe three types of possible asymptotic behaviour of these proportions in the case of non-unique population quantile. As an application of our findings we establish limiting properties of numbers of ties with a central order statistics in a discrete sample. Our results are derived not only for independent and identically distributed observations but more generally for strictly stationary and ergodic sequences of random variables.