Cesaro Sense Research Articles

Cesaro summation by spheres of lattice sums and Madelung constants

<p style='text-indent:20px;'>We study convergence of 3D lattice sums via expanding spheres. It is well-known that, in contrast to summation via expanding cubes, the expanding spheres method may lead to formally divergent series (this will be so e.g. for the classical NaCl-Madelung constant). In the present paper we prove that these series remain convergent in Cesaro sense. For the case of second order Cesaro summation, we present an elementary proof of convergence and the proof for first order Cesaro summation is more involved and is based on the Riemann localization for multi-dimensional Fourier series.</p>

Mean Convergence Theorems and Weak Laws of Large Numbers for Arrays of Measurable Operators under Some Conditions of Uniform Integrability

In this paper, we introduce the notions of uniform integrability in the Cesaro sense, h-integrability with respect to the array of constants {ani}, and h-integrability with exponent r for an array of measurable operators. Then, we establish some mean convergence theorems and weak laws of large numbers for arrays of measurable operators under some conditions related to these notions.

Laws of Large Numbers for Non-Homogeneous Markov Systems

In the present we establish Laws of Large Numbers for Non-Homogeneous Markov Systems and Cyclic Non-homogeneous Markov systems. We start with a theorem, where we establish, that for a NHMS under certain conditions, the fraction of time that a membership is in a certain state, asymptotically converges in mean square to the limit of the relative population structure of memberships in that state. We continue by proving a theorem which provides the conditions under which the mode of covergence is almost surely. We continue by proving under which conditions a Cyclic NHMS is Cesaro strongly ergodic. We then proceed to prove, that for a Cyclic NHMS under certain conditions the fraction of time that a membership is in a certain state, asymptotically converges in mean square to the limit of the relative population structure in the strongly Cesaro sense of memberships in that state. We then proceed to establish a founding Theorem, which provides the conditions under which, the relative population structure asymptotically converges in the strongly Cesaro sense with geometrical rate. This theorem is the basic instrument missing to prove, under what conditions the Law of Large Numbers for a Cycl-NHMS is with almost surely mode of convergence. Finally, we present two applications firstly for geriatric and stroke patients in a hospital and secondly for the population of students in a University system.

Probability distributions for Markov chain based quantum walks

We analyze the probability distributions of the quantum walks induced from Markov chains by Szegedy (2004). The first part of this paper is devoted to the quantum walks induced from finite state Markov chains. It is shown that the probability distribution on the states of the underlying Markov chain is always convergent in the Cesaro sense. In particular, we deduce that the limiting distribution is uniform if the transition matrix is symmetric. In the case of a non-symmetric Markov chain, we exemplify that the limiting distribution of the quantum walk is not necessarily identical with the stationary distribution of the underlying irreducible Markov chain. The Szegedy scheme can be extended to infinite state Markov chains (random walks). In the second part, we formulate the quantum walk induced from a lazy random walk on the line. We then obtain the weak limit of the quantum walk. It is noted that the current quantum walk appears to spread faster than its counterpart-quantum walk on the line driven by the Grover coin discussed in literature. The paper closes with an outlook on possible future directions.

Limiting measures for addition modulo a prime number cellular automata

Linear cellular automata have many invariant measures in general. There are several studies on their rigidity: The unique invariant measure with a suitable non-degeneracy condition (such as positive entropy or mixing property for the shift map) is the uniform measure - the most natural one. This is related to study of the asymptotic randomization property: Iterates starting from a large class of initial measures converge to the uniform measure (in Cesaro sense). In this paper we consider one-dimensional linear cellular automata with neighborhood of size two, and study limiting distributions starting from a class of shift-invariant probability measures. In the two-state case, we characterize when iterates by addition modulo 2 cellular automata starting from a convex combination of strong mixing probability measures can converge. This also gives all invariant measures inside the class of those probability measures. We can obtain a similar result for iterates by addition modulo an odd prime number cellular automata starting from strong mixing probability measures.

An elementary approach to asymptotic behavior in the Ces\`{a}ro sense and applications to the Laplace and Stieltjes transforms

We present an elementary approach to asymptotic behavior of generalized functions in the Cesaro sense. Our approach is based on Yosida's subspace of Mikusinski operators. Applications to Laplace and Stieltjes transforms are given.

The Central Limit Theorem for th-Order Nonhomogeneous Markov Information Source

We prove a central limit theorem for th-order nonhomogeneous Markov information source by using the martingale central limit theorem under the condition of convergence of transition probability matrices for nonhomogeneous Markov chain in Cesaro sense.

The prime number theorem for Beurling's generalized numbers. New cases

This paper provides new cases of the prime number theorem for Beurling’s generalized prime numbers. Let N be the distribution of a generalized number system and let π be the distribution of its primes. It is shown that N(x) = ax + O(x/ log x) (C), γ > 3/2, where (C) stands for the Cesaro sense, is sufficient for the prime number theorem to hold, i.e., π(x) ∼ x/ log x. The Cesaro asymptotic estimate explicitly means that ∫ x 1 N(t)− at t ( 1− t x )m dt = O ( x log x ) , for somem ∈ N. Therefore, it includes Beurling’s classical condition. We also show that under these conditions the Mobius function, associated to the generalized number system, has mean value equal to 0. The methods of this article are based on complex Tauberian theorems for local pseudo-function boundary behavior and arguments from the theory of asymptotic behavior of Schwartz distributions.

Homogenization of semilinear PDEs with discontinuous averaged coefficients

We study the asymptotic behavior of solutions of semilinear PDEs. Neither periodicity nor ergodicity will be assumed. On the other hand, we assume that the coecients have averages in the Cesaro sense. In such a case, the averaged coecients could be discontinuous. We use a probabilistic approach based on weak convergence of the associated backward stochastic dierential equation (BSDE) in the Jakubowski $S$-topology to derive the averaged PDE. However, since the averaged coecients are discontinuous, the classical viscosity is not dened for the averaged PDE. We then use the notion of solution introduced in [7]. The existence of $L_p$-viscosity to the averaged PDE is proved here by using BSDEs techniques.

Distributional Point Values and Convergence of Fourier Series and Integrals

In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier transforms in the following way: If $f\in\mathcal{S}^{\prime}\left(\mathbb{R}\right)$ and $x_{0}\in\mathbb{R}$ , and $\widehat{f}$ is locally integrable, then $f(x_{0})=\gamma\ $ distributionally if and only if there exists k such that $\frac{1}{2\pi}\lim_{x\rightarrow\infty}\int_{-x}^{ax} \hat{f}(t)e^{-ix_{0}t}\,\mathrm{d}t=\gamma\ \ (\mathrm{C},k)\,$ , for each a > 0, and similarly in the case when $\widehat{f}$ is a general distribution. Here $(\mathrm{C},k)$ means in the Cesaro sense. This result generalizes the characterization of Fourier series of distributions with a distributional point value given in [5] by $\lim_{x\rightarrow\infty}\sum_{-x\leq n\leq ax}a_{n}e^{inx_{0}}=\gamma\ (\mathrm{C},k)\,$ . We also show that under some extra conditions, as if the sequence $\left\{a_{n}\right\} _{n=-\infty}^{\infty}$ belongs to the space $l^{p}$ for some $p\in\lbrack1,\infty)$ and the tails satisfy the estimate $\sum_{\left\vert n\right\vert \geq N}^{\infty}\left\vert a_{n}\right\vert ^{p}=O\left(N^{1-p}\right) $ ,\ as $N\rightarrow\infty$ , the asymmetric partial sums\ converge to $\gamma$ . We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We apply these results to lacunary Fourier series of distributions.

Dynamic Cesaro-Wardrop equilibration in networks

We analyze a routing scheme for a broad class of networks which converges (in the Cesaro sense) with probability one to the set of approximate Cesaro-Wardrop equilibria, an extension of the notion of a Wardrop equilibrium. The network model allows for wireline networks where delays are caused by flows on links, as well as wireless networks, a primary motivation for us, where delays are caused by other flows in the vicinity of nodes. The routing algorithm is distributed, using only the local information about observed delays by the nodes, and is moreover impervious to clock offsets at nodes. The scheme is also fully asynchronous, since different iterates have their own counters and the orders of packets and their acknowledgments may be scrambled. Finally, the scheme is adaptive to the traffic patterns in the network. The demonstration of convergence in a fully dynamic context involves the treatment of two-time scale distributed asynchronous stochastic iterations. Using an ordinary differential equation approach, the invariant measures are identified. Due to a randomization feature in the algorithm, a direct stochastic analysis shows that the algorithm avoids non-Wardrop equilibria. Finally, some comments on the existence, uniqueness, stability, and other properties of Wardrop equilibria are made.

Characterization ofL p (R n ) using Gabor frames

We characterize L p norms of functions on R n for 1 < p < 1 in terms of their Gabor coecients. Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of L p functions converge to the functions almost everywhere and in L p for 1 < p < 1. In L 1 we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L 1 in a certain Cesaro sense. Consequently, we are able to establish that a large class of Gabor families generate Banach frames for L p (R n ) when 1 p < 1. It is a well known fact that certain Gabor expansions form frames for the Hilbert spaceL 2 (R n ). In this paper we show that that these expansions also generate Banach frames for L p (R n ) when 1 p < 1. The case p = 1 must be handled separately using Cesaro-Fejer summability (instead of Dirichlet summability) of partial sums. We also establish that certain Gabor expansions of L p functions converge to the functions almost everywhere and in L p (R n ) for 1 p < 1, where the convergence is interpreted in the Cesaro sense when p = 1. Moreover, we obtain frame inequali- ties, that characterize the L p (R n ) norm of f (up to equivalence) in terms of certain sequence-space norms evaluated at the sequence of Gabor coecients

Random Logistic Maps. I

Let {Ci}∞0 be a sequence of independent and identically distributed random variables with vales in [0, 4]. Let {Xn}∞0 be a sequence of random variables with values in [0, 1] defined recursively by Xn+1=Cn+1Xn(1−Xn). It is shown here that: (i) E ln C1 0, E |ln(4−C1)| 0 and −∫ ln(1−x) π(dx)=E ln C1. (v) E ln C1>0, E |ln(4−C1)|<∞ and {Xn} is Harris irreducible implies that the probability distribution of Xn converges in the Cesaro sense to a unique probability distribution on (0, 1) for all X0≠0.

Characterization of the Fourier series of a distribution having a value at a point

Let f be a periodic distribution of period 2π. Let ∑∞ n=−∞ ane inθ be its Fourier series. We show that the distributional point value f(θ0) exists and equals γ if and only if the partial sums ∑ −x≤n≤ax ane inθ0 converge to γ in the Cesaro sense as x→∞ for each a > 0.

A paradigm for class identification problems

The following problem arises in many applications involving classification, identification, and inference. There is a set of objects X, and a particular x /spl isin/ X is chosen (unknown to us). Based on information obtained about x in a sequential manner, one wishes to decide whether x belongs to one class of objects A/sub 0/ or a different class of objects A/sub 1/. The authors study a general paradigm applicable to a broad range of problems of this type, which they refer to as problems of class identification or discernibility. They consider various types of information sequences, and various success criteria including discernibility in the limit, discernibility with a stopping criterion, uniform discernibility, and discernibility in the Cesaro sense. They consider decision rules both with and without memory. Necessary and sufficient conditions for discernibility are provided for each case in terms of separability conditions on the sets A/sub 0/ and A/sub 1/. They then show that for any sets A/sub 0/ and A/sub 1/, various types of separability can be achieved by allowing failure on appropriate sets of small measure. Applications to problems in language identification, system identification, and discrete geometry are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An infinite-dimensional law of large numbers in Cesaro's sense

Let (X k) k≥0 be a sequence of independent copies of a random variableX taking its values in a real separable Banach space (B, ¦ ¦). For every real number β>−1 one defines the following coefficients: $$A_0^\beta = 1, A_1^\beta = \beta + 1,..., A_k^\beta = (\beta + 1) \cdots (\beta + k)/k!,...$$ It is shown that for all α∈]0, 1[ the sequenceV n =(1/A α )∑0⩽k⩽n A α−1 X k converges almost surely toE(X) if and only if ‖X‖1/α is integrable. This extends results obtained earlier by several authors for scalar-valued random variables: Lorentz (case 1/2<α<1), Chow and Lai (case 0<α<1/2), Deniel and Derriennic (case α=1/2).

Direct Adaptive Control with the Robust Least Squares Method

An adaptive control of linear dynamic single input-single output systems des-cribed by the ARMAX model in the presence of uncertainty is considered. An adaptive control algorithm is derived using the Huber’s min-max robust estimation, based on a priori information of disturbance statistics, such as the pdf’s class to which the unknown disturbance pdf belongs. The derived algorithm differs from the least squares method by the insertion of an optimal nonlinear transformation of the prediction error within a given class, in the sense of the minimal Cramer Rao bound. Moreover, the algorithm possesses a high convergence rate at initial steps, owing to a suitable way of the recursive generation of its weighting matrix, which is adaptial to the chosen non-linearity. The stability, in the Cesaro sense, of the proposed control system and the asymptotic optimality of the adaptive regulator is established theoretically using the martingale theory. It is worthwile noting that the obtained results are valid under weaker constrains on the disturbance polynomial than in the case of convenient least squares method.

A Robust Adaptive Minimum Variance Controller

This paper addresses the twin questions of performance and robustness of an adaptive controller for single-input, single-output, linear, stochastic systems. The authors present an adaptive controller that has the following properties: (1) Attaining optimal regulation and tracking in the ideal, minimum phase, known upper bound on system order, known sign and lower bound for the leading coefficient $(b_0 )$, positive real condition on noise case, and self-tuning in a Cesaro sense to a minimum variance regulator in the case of pure regulation. (2) Providing mean square stability when the system is of minimum phase, with known upper bound on order but not necessarily satisfying a positive real condition on the noise. (3) Providing mean square stability when the system is in a graph topological neighborhood (of computable size) of an ideal plant as in (1), and the statistical properties of the disturbance are violated. (4) Continuing to stabilize the system when the adaptation gain is prevented from vanishing.

On the convergence of self-tuning stochastic servo algorithms based on stochastic approximation

The problem of the adaptive tracking of stochastic reference signals is considered, supposing that the process can be represented by an ARMAX model with arbitrary time delay and the reference signal by an ARMA model. Two algorithms of the stochastic approximation type, providing adaptation to both process and reference characteristics, are proposed. They differ by the incorporated a priori knowledge about the optimal regulator parameters. Global stability, asymptotic optimally, convergence of the adaptive control law in a Cesaro sense, and the strong consistency of the parameter estimates are proved. The persistence-of-excitation condition is analyzed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Adaptive control with the stochastic approximation algorithm: Geometry and convergence

New geometric properties possessed by the sequence of parameter estimates are exhibited, which yield valuable insight into the behavior of the stochastic approximation based algorithm as it is used in minimum variance adaptive control. In particular, these geometric properties, together with certain probabilistic arguments, prove that if the system does not have a reduced-order minimum variance controller, then the parameter estimates converge to a random multiple of the true parameter. An explicit expression for the limiting parameter estimate is also available. With strictly positive probability, the limiting parameter estimate is not the true parameter, and in some cases differs from the true parameter with probability one. If the system possesses reduced-order minimum variance controllers, then convergence to a minimum variance controller in a Cesaro sense is shown. The geometry of the limit set is described. Sufficient conditions are also given for some of these results to hold for parameter estimation schemes other than stochastic approximation.