Finite Second-order Moments Research Articles

Approximation of Second-Order Moment Processes from Local Averages

We use local averages to approximate processes that have finite second-order moments and are continuous in quadratic mean. We also provide some insight and generalization of the connection between Bernstein polynomials and Brownian motion, which was investigated by Kowalski in 2006.

Euler–Poisson Systems as Action-Minimizing Paths in the Wasserstein Space

This paper uses a variational approach to establish existence of solutions (σt, vt) for the 1-d Euler–Poisson system by minimizing an action. We assume that the initial and terminal points σ0, σT are prescribed in \({\mathcal {P}_2(\mathbb {R})}\) , the set of Borel probability measures on the real line, of finite second-order moments. We show existence of a unique minimizer of the action when the time interval [0,T] satisfies T < π. These solutions conserve the Hamiltonian and they yield a path t → σt in \({\mathcal {P}_2(\mathbb {R})}\) . When σt = δy(t) is a Dirac mass, the Euler–Poisson system reduces to \({\ddot {y} + y=0}\) . The kinetic version of the Euler–Poisson, that is the Vlasov–Poisson system was studied in Ambrosio and Gangbo (Comm Pure Appl Math, to appear) as a Hamiltonian system.

Pressureless Euler/Euler–Poisson Systems via Adhesion Dynamics and Scalar Conservation Laws

The “sticky particles” model at the discrete level is employed to obtain global solutions for a class of systems of conservation laws among which lie the pressureless Euler and the pressureless attractive/repulsive Euler–Poisson system with zero background charge. We consider the case of finite, nonnegative initial Borel measures with finite second-order moment, along with continuous initial velocities of at most quadratic growth and finite energy. We prove the time regularity of the solution for the pressureless Euler system and obtain that the velocity satisfies the Oleinik entropy condition, which leads to a partial result on uniqueness. Our approach is motivated by earlier work of Brenier and Grenier, who showed that one-dimensional conservation laws with special initial conditions and fluxes are appropriate for studying the pressureless Euler system.

Adaptive filtering algoritm for time-variant symmetric α-stable signals

The existence of non-Gaussian distributed impulsive signals has roots in the generalized central theorem. In particular, the symmetric α-stable (SαS) distribution has been used to model heavy-tailed phenomena encountered in communication, underwater acoustics, and radar. Most previous work in the field of active control uses some of the numerous variants of LMS algorithm extended with the filtered-x scheme. The stable processes, however, do not possess finite second-order moments. Hence, the filter should be based on lp-norm optimization. The p-norm enters the expression for the tap-weight update [p(t)&amp;lt;α̂(t)]. In this work a new adaptive algorithm referred to as regularized normalized least−mean p−norm algorithm [εNLMp(t)] is presented. A running estimate α̂(t) of the time−variant characteristic exponent of the stable signal is obtained. Hence, by adaptively tuning the filter p−norm a more optimal filter performance results in time−variant situations. Simulation analysis provides insight into the adaptive filter performance for various time−invariant and time−variant noisy signals for 1&amp;lt;α(t)≤2. Moreover, it is demonstrated that the filter performance is only marginally degraded compared with LMS algorithm for ordinary time−invariant Gaussian noise signals. Performance of an active control system exposed to time−varying heavy−tailed noise signals is presented.

Some Results on the Self-Similarity Property in Communication Networks

Due to the strong experimental evidence that packet network traffic is self-similar in nature, it is important to study the problems to see whether the superposition of self-similar processes retains the property of self-similarity, and whether the service of a server changes the self-similarity property of the input traffic. In this letter, we first discuss some definitions and superposition properties of self-similar processes. We obtain some good results about the property of merging self-similar data streams. Then we present a model of a single server with infinite buffer and prove that when the queue length has finite second-order moment, the input process, being strong asymptotically second-order self-similar (sas-s), is equivalent to the output process which also bears the sas-s property.

Asymptotic Chi-Square Tests for a Large Class of Factor Analysis Models

Three types of asymptotic $\chi^2$ goodness-of-fit tests derived under the normal assumption have been used widely in factor analysis. Asymptotic behavior of the test statistics is investigated here for the factor analysis model with linearly or nonlinearly restricted factor loadings under weak assumptions on the factor vector and the error vector. In particular the limiting $\chi^2$ result for the three tests is shown to hold for the factor vector, either fixed or random with any distribution having finite second-order moments, and for the error vector with any distribution having finite second-order moments, provided that the components of the error vector are independent, not just uncorrelated. As special cases the result holds for exploratory and confirmatory factor analysis models and for certain nonnormal structural equation (LISREL) models.

Decentralized multicriteria optimization of linear stochastic systems

By adopting a decision-theoretic approach and under the noncooperative equilibrium solution concept of game theory, decentralized multicriteria optimization of stochastic linear systems with quasiclassical information patterns is discussed. First, the static M -person quadratic decision problem is considered, and sufficiency conditions are derived for existence of a unique equilibrium solution when the primitive random variables have a priori known but arbitrary probability distributions with finite second-order moments. The optimal strategies are given in the form of the limit of a convergent sequence which is shown to admit a closed-form linear solution for the special case of Gaussian distributions. Then, this result is generalized to dynamic LQG problems, and a general theorem is proven, which states that under the one-step-delay observation sharing pattern this class of systems admit unique affine equilibrium solutions. This result, however, no longer holds true under the one-step-delay sharing pattern, and additional criteria have to be introduced in this case. These results are then interpreted within the context of LQG team problems, so as to generalize and unify some of the results found in the literature on team problems.

Suboptimal controls of a second-order system subjected to discontinuous random perturbations

This paper doala with tho suboptimal control of a stochastic second-order system. It is assumed that the system is subjected to two different kinds of perturbation. The first kind of perturbation is represented by the standard Wiener process and tho second kind by a homogeneous process with independent increments, finite second-order moments, mean zero and no continuous sample paths. The stochastic system is approximated by a diffusion type system, and the optimal feedback control laws for the diffusion type system are chosen as the suboptimal controls for the given system. It is shown that in order to construct tho suboptimal controls, a non-linear partial differential equation and a linear partial integro-difforontial equation, have to be solved. A finite difference algorithm for the solution of these equations is proposed, and its efficiency and applicability are demonstrated with examples.

On the Computation of Functionals Over Sample Paths of a Randomly Rotating Rigid Body

This paper deals with the computation of the values of two functionals which are defined over the sample paths of a randomly rotating rigid body. It is assumed that the body is subjected to two different kinds of perturbation. The first kind of perturbation is represented by the standard Wiener process and the second kind by a homogeneous process with independent increments, finite second-order moments, mean zero and no continuous sample functions. In order to measure quantitatively the stochastic stability of the body's motion, two functionals are defined over its sample paths. It is shown that each of these functionals is a solution to a corresponding partial integro-differential equation. A numerical procedure for the solution of these equations is suggested, and its efficiency and applicability are demonstrated with examples.

Equilibrium solutions in two-person quadratic decision problems with static information structures

This paper is concerned with the class of two-person two-objective decision problems characterized by quadratic cost functions and static information structures. The primitive random variables are assumed to have a priori known but arbitrary probability, distributions with finite second-order moments. For this class of problems, sufficient conditions are derived for the existence of a unique pair of equilibrium solutions. These sufficient conditions are independent of the probabilistic structure of the problem. When the underlying probability, distribution is Gaussian and the observations of the decision makers are linear in the primitive random variables, then it is shown that these unique equilibrium solutions are linear and can be found as the unique solution of a Lyapunov-type matrix equation. For the two special extreme cases known as minimax and team problems, these linear policies constitute a saddle-point solution and a globally-optimal team solution, respectively-thus being in agreement with the existing results in the literature for these two extreme cases.