Inequalities In Probability Theory Research Articles

Cooperative Spacing Sampled Control of Vehicle Platoon Considering Undirected Topology and Analog Fading Networks

The work on this paper studies the combined effect of vehicle dynamics, sampling period and the quality of inter-vehicle sensing and communication on the performance of platoon control. First, we induce the sampled control protocol from continuous–time linear consensus protocol by employing zero–order hold circuit and periodic sampling technology, then by virtue of the obtained sampled control protocol, the continuous–time platoon system is equivalently transformed into a discrete–time system. Second, for the inter–vehicle communication with ideal channel and under undirected information flow topology (UIFT), the stability thresholds of control gains are explicitly established by solving a discrete–time simultaneous stabilization problem, bilinear transformation and the Routh-Hurwitz stability criterion. Meanwhile, to ensure the fastest asymptotic stability of the platoon dynamics, we propose an optimal asymptotic convergence factor and a correspondingly optimal sampled control protocol. Third, For the inter–vehicle communication with identical fading networks and under UIFT, through solving a modified Riccati inequality, we provide a sampled control protocol depending not only on information flow topology (IFT) and sampling period, but also on the statistics of the inter–vehicle communication channel. Besides, employing Lyapunov inequality in probability theory, we show a necessary condition for the sampled control protocol ensuring the platoon dynamics mean square stable. Finally, simulations are performed to demonstrate the theory’s discoveries.

Integral inequalities in probability theory revisited

In [1], the following conjecture was proposed concerning the distribution of ages in a closed interval [0, A]

Defining the threshold bounds for effective modelling of aggregated IDS

The idea behind the aggregation of multiple intrusion detections (IDS) is due to the inability of stand-alone IDS to meet the demand of today’s network traffic with high complexity of intrusions. Multiple IDS can be aggregated in order to meet the requirement for low false alarms and the demand for high detection rate. This paper shows that by appropriate adjustment in the threshold results in better performance during the aggregation process of the multiple IDS. Complementary IDS are used to increase the detection rate rather than the individual IDS. A number of evaluation metrics is used to show that there is a significant enhancement in the performance of the IDS system. The threshold bounds are derived using three well known inequalities in probability theory and compared for better performance.

Zeta Functions and the Log Behaviour of Combinatorial Sequences

AbstractIn this paper, we use the Riemann zeta functionζ(x) and the Bessel zeta functionζμ(x) to study the log behaviour of combinatorial sequences. We prove thatζ(x) is log-convex forx> 1. As a consequence, we deduce that the sequence {|B2n|/(2n)!}n≥ 1 is log-convex, whereBnis thenth Bernoulli number. We introduce the functionθ(x) = (2ζ(x)Γ(x + 1))1/x, whereΓ(x)is the gamma function, and we show that logθ(x) is strictly increasing forx≥ 6. This confirms a conjecture of Sun stating that the sequenceis strictly increasing. Amdeberhanet al. defined the numbersan(μ)= 22n+1(n+ 1)!(μ+ 1)nζμ(2n) and conjectured that the sequence{an(μ)}n≥1is log-convex forμ= 0 andμ= 1. By proving thatζμ(x)is log-convex forx >1 andμ >-1, we show that the sequence{an(≥)}n>1 is log-convex for anyμ >- 1. We introduce another functionθμ,(x)involvingζμ(x)and the gamma functionΓ(x)and we show that logθμ(x)is strictly increasing forx >8e(μ+ 2)2. This implies thatBased on Dobinski’s formula, we prove thatwhereBnis thenth Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property ofand Holder’s inequality in probability theory.

An elementary proof of the covariance inequality for Choquet integral

The covariance inequality, also known as Chebyshev’s inequality, is a very important inequality in probability theory. In generalized measure theory, the proof of this inequality for Choquet integral has a long process. In this paper, we present a simple proof of Chebyshev’s inequality for Choquet integral which requires only knowledge of some basic properties of Choquet integral.

95.20 Integral inequalities in probability theory

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A Semidefinite Programming Approach to Optimal-Moment Bounds for Convex Classes of Distributions

We provide an optimization framework for computing optimal upper and lower bounds on functional expectations of distributions with special properties, given moment constraints. Bertsimas and Popescu (Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 2004. Forthcoming) have already shown how to obtain optimal moment inequalities for arbitrary distributions via semidefinite programming. These bounds are not sharp if the underlying distributions possess additional structural properties, including symmetry, unimodality, convexity, or smoothness. For convex distribution classes that are in some sense generated by an appropriate parametric family, we use conic duality to show how optimal moment bounds can be efficiently computed as semidefinite programs. In particular, we obtain generalizations of Chebyshev’s inequality for symmetric and unimodal distributions and provide numerical calculations to compare these bounds, given higher-order moments. We also extend these results for multivariate distributions.

A Conic Programming Approach to Generalized Tchebycheff Inequalities

Consider the problem of finding optimal bounds on the expected value of piecewise polynomials over all measures with a given set of moments. This is a special class of generalized Tchebycheff inequalities in probability theory. We study this problem within the framework of conic programming. Relying on a general approximation scheme for conic programming, we show that these bounds can be numerically computed or approximated via semidefinite programming. We also describe some applications of this class of generalized Tchebycheff inequalities in probability, finance, and inventory theory.

Optimal Inequalities in Probability Theory: A Convex Optimization Approach

We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for $P(X\in S)$, for a set S defined by polynomial inequalities and a random vector X defined on $\Omega\subseteq {\cal R}^n$ that has a given collection of up to kth-order moments. In the univariate case, we provide optimal bounds on $P(X\in S)$, when the first k moments of X are given, as the solution of a semidefinite optimization problem in k + 1 dimensions. In the multivariate case, if the sets S and $\Omega$ are given by polynomial inequalities, we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n, for fixed k. We characterize the complexity of the problem of deriving tight moment inequalities. We show that it is NP-hard to find tight bounds for $k\geq 4$ and $\Omega={\cal R}^n$ and for $k\geq 2$ and $\Omega={\cal R}^n_+,$ when the data in the problem is rational. For k = 1 and $\Omega={\cal R}^n_+$ we show that we can find tight upper bounds by solving n convex optimization problems when the set S is convex, and we provide a polynomial time algorithm when S and $\Omega$ are unions of convex sets, over which linear functions can be optimized efficiently. For the case k = 2 and $\Omega={\cal R}^n$, we present an efficient algorithm for finding tight bounds when S is a union of convex sets, over which convex quadratic functions can be optimized efficiently.

Inequalities in probability theory and turán-type problems for graphs with colored vertices

The “best” inequalities of type P{(ζ, η)⊂ E} ≧f(P{η⊂ D1}P{η⊂Dm}) for independent and identically distributed random elements ζ and η can be reduced to Turan-type problems for graphs with colored vertices. In the present work we describe a finite algorithm for obtaining the asymptotical solution for an arbitrary problem of such type. In the case of two colors we obtain the final form of asymptotic solution without using the algorithm.

An Inequality in Probability Theory

An Inequality in Probability Theory

An inequality in probability theory

1. Let x and y be random variables with finite expectations. We shall say that x dominates y if e I{(x) } > E {+(y) } whenever q5 is a continuous convex function on the real line R1. (The expectations ?{+(x) } and ?{+(y) } are always well defined if + oo is admitted as a value.) Assume now that xi and x2 are independent and dominate respectively the independent random variables yi and Y2. Let q5 be a continuous convex function on R2 and denote by Fi and Gi the distribution functions of xi and yi. We have