Inequalities For Sums Research Articles

A Small Delay and Correlation Time Limit of Stochastic Differential Delay Equations with State-Dependent Colored Noise

We consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. An Ornstein-Uhlenbeck process is used to model the colored noise. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.

The isotropic constant of random polytopes with vertices on convex surfaces

For an isotropic convex body K⊂Rn we consider the isotropic constant LKN of the symmetric random polytope KN generated by N independent random points which are distributed according to the cone probability measure on the boundary of K. We show that with overwhelming probability LKN≤Clog(2N∕n), where C∈(0,∞) is an absolute constant. If K is unconditional we argue that even LKN≤C with overwhelming probability and thereby verify the hyperplane conjecture for this model. The proofs are based on concentration inequalities for sums of sub-exponential or sub-Gaussian random variables, respectively, and, in the unconditional case, on a new ψ2-estimate for linear functionals with respect to the cone measure in the spirit of Bobkov and Nazarov, which might be of independent interest.

New Stability Criteria of Discrete Systems With Time-Varying Delays

This paper is concerned with the stability criteria of discrete neural networks with two additive input time-varying delays. By using the time delay division and a new summation inequality, a less conservative criterion is derived. Moreover, to compare the obtained criterion more directly with the existing results, a corollary is proposed accordingly. Finally, some numerical examples are presented to demonstrate that the obtained criteria are less conservative.

An Insight into Stability Conditions of Discrete-Time Systems With Time-Varying Delay

This note studies an interesting phenomenon for stability conditions of discrete-time systems with time-varying delay. The underlying reason behind this phenomenon is revealed, and thereafter some conclusions are drawn: (i) Stability conditions of discrete-time systems with time-varying delay are generally divided into two types: those obtained by summation inequalities with free-matrix variables and those obtained by the combination of summation inequalities without free-matrix variables and the reciprocally convex lemma; (ii) The conservatism between the two types of stability conditions can not be theoretically compared. To clearly demonstrate this interesting phenomenon and meanwhile, to further verify these conclusions, several bounded real lemmas are obtained via different bounding-inequality methods and applied to a numerical example.

Generalizations of numerical radius inequalities related to block matrices

We establish several numerical radius inequalities related to 2x2 positive semidefinite block matrices. It is shown that if A,B,C ? Mn(C) are such that [A B* B C] ? 0, then wr(B)? 1/2 ||Ar + Cr||, for r ? 1. Related numerical radius inequalities for sums and products of matrices are also given.

Numerical Radius Inequalities for Sums and Products of Operators

A numerical radius inequality due to Shebrawi and Albadawi says that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then for all r≥1. We give sharper numerical radius inequality which states that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then where . Moreover, we give many numerical radius inequalities which are sharper than related inequalities proved recently, and several applications are given.

Probability inequalities for sums of WUOD random variables and their applications

Probability inequalities for sums of WUOD random variables and their applications

A Novel Approach to Delay-Variation-Dependent Stability Analysis of 2-D Discrete-Time Systems With Mixed Delays

In this paper, a stability analysis problem is studied for a class of two-dimensional (2-D) discrete-time systems with time-varying and distributed delays described by the second Fornasini-Marchesini (FM) model. First, new 2-D polynomials-based summation inequalities are proposed to estimate summation terms in the forward difference of Lyapunov-Krasovskii functional (LKF). The inequalities can reduce to 2-D Jensen inequalities and 2-D finite-sum inequalities by designing slack matrices and arbitrary vectors. Second, a new augment LKF is constructed, which makes full use of the delay changing information. By the Lyapunov stability theory, sufficient conditions for asymptotic stability of 2-D discrete-time systems are derived in the form of linear matrix inequalities. Finally, two simulation examples are given to demonstrate the effectiveness of the proposed methods.

Probability inequalities for sums of NSD random variables and applications

In this paper, we obtain the exponential-type inequalities for maximal partial sums of negatively superadditive dependent (NSD) random variables, which extends the corresponding results for independent and negatively associated (NA) random variables. Using these inequalities, we further investigate the weak convergence of the M-estimators in the generalized linear model with NSD errors, which generalize and improve the corresponding results of the independent random errors to that of NSD random errors.

Bessel summation inequalities for stability analysis of discrete‐time systems with time‐varying delays

SummaryThis paper is concerned with the stability analysis problems of discrete‐time systems with time‐varying delays using summation inequalities. In the literature focusing on the Lyapunov‐Krasovskii approach, the Jensen integral/summation inequalities have played important roles to develop less conservative stability criteria and thus have been widely studied. Recently, the Jensen integral inequality was successfully generalized to the Bessel‐Legendre inequalities constructed with arbitrary‐order Legendre polynomials. It was also shown that general inequality contributes to the less conservatism of stability criteria. In the case of discrete‐time systems, however, the Jensen summation inequality are hardly extensible to general ones since there have still not been general discrete orthogonal polynomials applicable to the developments of summation inequalities. Motivated by such observations, this paper proposes novel discrete orthogonal polynomials and then successfully derives general summation inequalities. The resulting summation inequalities are discrete‐time counterparts of the Bessel‐Legendre inequalities but are not based on the discrete Legendre polynomials. By developing hierarchical stability criteria based on the proposed summation inequalities, the effectiveness of the proposed approaches is demonstrated via three numerical examples for the stability analysis of discrete‐time systems with time‐varying delays.

Inference on Causal and Structural Parameters using Many Moment Inequalities

Abstract This article considers the problem of testing many moment inequalities where the number of moment inequalities, denoted by $p$, is possibly much larger than the sample size $n$. There is a variety of economic applications where solving this problem allows to carry out inference on causal and structural parameters; a notable example is the market structure model of Ciliberto and Tamer (2009) where $p=2^{m+1}$ with $m$ being the number of firms that could possibly enter the market. We consider the test statistic given by the maximum of $p$ Studentized (or $t$-type) inequality-specific statistics, and analyse various ways to compute critical values for the test statistic. Specifically, we consider critical values based upon (1) the union bound combined with a moderate deviation inequality for self-normalized sums, (2) the multiplier and empirical bootstraps, and (3) two-step and three-step variants of (1) and (2) by incorporating the selection of uninformative inequalities that are far from being binding and a novel selection of weakly informative inequalities that are potentially binding but do not provide first-order information. We prove validity of these methods, showing that under mild conditions, they lead to tests with the error in size decreasing polynomially in $n$ while allowing for $p$ being much larger than $n$; indeed $p$ can be of order $\exp (n^{c})$ for some $c > 0$. Importantly, all these results hold without any restriction on the correlation structure between $p$ Studentized statistics, and also hold uniformly with respect to suitably large classes of underlying distributions. Moreover, in the online supplement, we show validity of a test based on the block multiplier bootstrap in the case of dependent data under some general mixing conditions.

A Combinatorial Approach to Small Ball Inequalities for Sums and Differences

Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent, identically distributed random elements taking values in very general sets. Depending on the setting – abelian or non-abelian groups, or vector spaces, or Banach spaces – we provide a collection of inequalities relating different small ball probabilities that are sharp in many cases of interest. We prove these distribution-free probabilistic inequalities by showing that underlying them are inequalities of extremal combinatorial nature, related among other things to classical packing problems such as the kissing number problem. Applications are given to moment inequalities.

Stability analysis of discrete-time neural networks with an interval-like time-varying delay

This paper studies the stability of discrete-time neural networks with an interval-like time-varying delay via the Lyapunov–Krasovskii (L–K) functional method. Firstly, a general free-matrix-based (GFMB) summation inequality is proposed, which includes the auxiliary-function-based and conventional free-matrix-based ones. Secondly, a novel L–K functional is deliberately constructed to make full advantage of the newly-developed GFMB summation inequality. Finally, a new stability condition is derived and two widely-used numerical examples are presented to show the effectiveness of the proposed approach.

Robust H∞ Performance of Discrete-time Neural Networks with Uncertainty and Time-varying Delay

In this paper, we are concerned with the robust H∞ problem for a class of discrete-time neural networks with uncertainties. Under a weak assumption on the activation functional, some novel summation inequality techniques and using a new Lyapunov-Krasovskii (L-K) functional, a delay-dependent condition guaranteeing the robust asymptotically stability of the concerned neural networks is obtained in terms of a Linear Matrix Inequality(LMI). It is shown that this stability condition is less conservative than some previous ones in the literature. The controller gains can be derived by solving a set of LMIs. Finally, two numerical examples result are given to illustrate the effectiveness of the developed theoretical results.

Improved Stability Criteria for Discrete-time Delay Systems via Novel Summation Inequalities

This paper is concerned with the stability analysis of linear discrete time-delay systems. New discrete inequalities for single summation and double summation are presented to estimate summation terms in the forward difference of Lyapunov-Krasovskii functional (LKF), which are more general than some commonly used summation inequalities. Through the construction of an augmented LKF, improved delay-dependent stability criteria for discrete time-delay systems are established. Based on this, a time-delayed controller is derived for linear discrete time-delay systems. Finally, the advantages of the proposed criteria are revealed from the solutions of the numerical examples.

On Young's inequality

We present some inequalities for trigonometric sums. Among others, we prove the following refinements of the classical Young inequality.(1)Let m≥3 be an odd integer, then for all n≥m−1,∑k=1ncos⁡(kθ)k≥∑k=1m(−1)kk. The sign of equality holds if and only if n=m and θ=π. The special case m=3 is due to Brown and Koumandos (1997).(2)For all even integers n≥2 and real numbers r∈(0,1] and θ∈[0,π] we have∑k=1ncos⁡(kθ)krk≥−548(5+5)=−0.75375.... The sign of equality holds if and only if n=4, r=1 and θ=4π/5. We apply this result to prove the absolute monotonicity of a function which is defined in terms of the log-function.

KETAKSAMAAN JUMLAHAN SINUS PANGKAT 2^n YANG BERLAKU PADA SEGITIGA LANCIP

Let \alpha, \beta, \gamma are angles in acute triangle ABC and a,b,c are the length of the triangle. By using the sine of angles as the relationship between the length of triangle and the radius of the circle circumscribed about a plane triangle, will be proven the sum inequality of quadratic sine in acute triangle. Then, by using the quadratic sum inequality of the sides of triangle will be extended for the case of the sum inequality of sine of order 2^n in acute triangle.

Finite-Time Passivity-Based Stability Criteria for Delayed Discrete-Time Neural Networks via New Weighted Summation Inequalities.

In this paper, we study the problem of finite-time stability and passivity criteria for discrete-time neural networks (DNNs) with variable delays. The main objective is how to effectively evaluate the finite-time passivity conditions for NNs. To achieve this, some new weighted summation inequalities are proposed for application to a finite-sum term appearing in the forward difference of a novel Lyapunov-Krasovskii functional, which helps to ensure that the considered delayed DNN is passive. The derived passivity criteria are presented in terms of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed results.

Enhanced result on stability analysis of randomly occurring uncertain parameters, leakage, and impulsive BAM neural networks with time‐varying delays: Discrete‐time case

SummaryIn real‐world problems, neural networks play an increasingly important role in terms of both theory and applications. In this paper, the asymptotic stability analysis issue is investigated for uncertain impulsive discrete‐time bidirectional associative memory neural networks with leakage and time‐varying delays. With the assistance of novel summation inequality, reciprocally convex combination technique, and triple Lyapunov‐Krasovskii functionals terms, many cases of time‐varying delays are examined to certify the stability of neural networks. Here, the uncertainties are considered as a randomly occurring parameter uncertainty, and it conforms certain mutually uncorrelated Bernoulli‐distributed white noise sequences. An important feature of the results reported here is that the probability of occurrence of the parameter uncertainties specify a priori estimate. Finally, numerical examples are proposed to expose the capability and efficiency of our research work with the help of the LMI control toolbox in MATLAB.

An improvement of a non-uniform bound for combinatorial central limit theorem

ABSTRACTIn this article, we develop a new Rosenthal Inequality for uniform random permutation sums of random variables with finite third moments and apply it to obtain a sharp non-uniform bound for the combinatorial central limit theorem using the Stein's method and the exchangeable pair techniques. The obtained bound is shown to be sharper than other existing bounds.