Probability Inequalities Research Articles

English

The Gaussian correlation inequality for multivariate zero-mean normal probabilities of symmetrical n-rectangles can be considered as an inequality for multivariate gamma distributions (in the sense of Krishnamoorthy and Parthasarathy [5]) with one degree of freedom. Its generalization to all integer of and sufficiently large non-integer degrees of freedom was recently proved in [10]. Here, this inequality is partly extended to smaller non-integer of and in particular - in a weaker form - to all infinitely divisible multivariate gamma distributions. A further monotonicity property - sometimes called more PLOD (positively lower orthant dependent) - for increasing correlations is proved for multivariate gamma distributions with integer or sufficiently large of freedom.

Generalizations of some probability inequalities and $L^{p}$ convergence of random variables for any monotone measure

This paper has three specific aims. First, some probability inequalities, including Holder’s inequality, Lyapunov’s inequality, Minkowski’s inequality, concentration inequalities and Fatou’s lemma for Choquet-like expectation based on a monotone measure are shown, extending previous work of many researchers. Second, we generalize some theorems about the convergence of sequences of random variables on monotone measure spaces for Choquet-like expectation. Third, we extend the concept of uniform integrability for Choquet-like expectation. These results are useful for the solution of various problems in machine learning and made it possible to derive new efficient algorithms in any monotone system. Corresponding results are valid for capacities, the usefulness of which has been demonstrated by the rapidly expanding literature on generalized probability theory.

Euclidean Networks with a Backbone and a Limit Theorem for Minimum Spanning Caterpillars

A caterpillar network (or graph) G is a tree with the property that removal of the leaf edges of G leaves one with a path. Here we focus on minimum weight spanning caterpillars where the vertices are points in the Euclidean plane and the costs of the path edges and the leaf edges are multiples of their corresponding Euclidean lengths. The flexibility in choosing the weight for path edges versus the weight for leaf edges gives some useful flexibility in modeling. In particular, one can accommodate problems motivated by communications theory such as the “last mile problem.” Geometric and probabilistic inequalities are developed that lead to a limit theorem that is analogous to the well-known Beardwood, Halton, and Hammersley theorem for the length of the shortest tour through a random sample, but the minimal spanning caterpillars fall outside the scope of the theory of subadditive Euclidean functionals.

Sharp probability inequalities for reliable evacuation planning

This paper contributes to the literature by providing new techniques to determine optimal evacuation routes when limited information is known about the evacuation demand and road capacities. Specifically, this research complements prior work by requiring only their first and second moments to be known. The inclusion of an optional symmetry assumption is rigorously shown to further reduce the evacuation time. Finally, we demonstrate that the probability inequalities that serve to ensure the confidence level of the evacuation plan are sharp. A case study illustrates the core idea of demand inflation and supply deflation in obtaining reliable evacuation plans.

New Analysis on Sparse Solutions to Random Standard Quadratic Optimization Problems and Extensions

The standard quadratic optimization problem (StQP) refers to the problem of minimizing a quadratic form over the standard simplex. Such a problem arises from numerous applications and is known to be NP-hard. In a recent paper [Chen X, Peng J, Zhang S (2013) Sparse solutions to random standard quadratic optimization problems. Math. Programming 141(1–2):273–293], Chen, et al. showed that with a high probability close to 1, StQPs with random data have sparse optimal solutions when the associated data matrix is randomly generated from a certain distribution such as uniform and exponential distributions. In this paper, we present a new analysis for random StQPs combining probability inequalities derived from the first-order and second-order optimality conditions. The new analysis allows us to significantly improve the probability bounds. More important, it allows us to handle normal distributions, which is left open in Chen et al. (2013). The existence of sparse approximate solutions to convex StQPs and extensions to other classes of QPs are discussed as well.

Tight Probabilistic SINR Constrained Beamforming Under Channel Uncertainties

In downlink multi-user beamforming, a single basestation is serving a number of users simultaneously. However, energy intended for one user may leak to other unintended users, causing interference. With signal-to-interference-plus-noise ratio (SINR) being one of the most crucial quality metrics to users, beamforming design with SINR guarantee has always been an important research topic. However, when the channel state information is not accurate, the SINR requirements become probabilistic constraints, which unfortunately are not tractable analytically for general uncertainty distribution. Therefore, existing probabilistic beamforming methods focus on the relatively simple Gaussian and uniform channel uncertainties, and mainly rely on probability inequality based approximated solutions, resulting in conservative SINR outage realizations. In this paper, based on the local structure of the feasible set in the probabilistic beamforming problem, a systematic method is proposed to realize tight SINR outage control for a large class of channel uncertainty distributions. With channel estimation and quantization errors as examples, simulation results show that the SINR outage can be realized tightly, which results in reduced transmit power compared to the existing inequality based probabilistic beamformers.

Exponential probability inequalities for WNOD random variables and their applications

Some exponential probability inequalities for widely negative orthant dependent (WNOD, in short) random variables are established, which can be treated as very important roles to prove the strong law of large numbers among others in probability theory and mathematical statistics. By using the exponential probability inequalities, we study the complete convergence for arrays of rowwise WNOD random variables. As an application, the Marcinkiewicz–Zygmund type strong law of large numbers is obtained. In addition, the complete moment convergence for arrays of rowwise WNOD random variables is studied by using the exponential probability inequality and complete convergence that we established.

Exponential probability inequality for $$m$$ m -END random variables and its applications

The concept of \(m\)-extended negatively dependent (\(m\)-END, in short) random variables is introduced and the Kolmogorov exponential inequality for \(m\)-END random variables is established. As applications of the Kolmogorov exponential inequality, we further investigate the complete convergence for arrays of rowwise \(m\)-END random variables and the complete consistency for the estimator of nonparametric regression models based on \(m\)-END errors. Our results generalize and improve some known ones for independent random variables and dependent random variables.

On a probability inequality of van Dam

This note derives an interesting probability inequality between the expectation of a conditional variance and the variance of a conditional expectation for a function of two independent random variables. In the special case of finite discrete random variables, the inequality coincides with an inequality by Feng and Tonge (2000), which extends a result by van Dam (1998).

Online and Non-Parametric Drift Detection Methods Based on Hoeffding’s Bounds

Incremental and online learning algorithms are more relevant in the data mining context because of the increasing necessity to process data streams. In this context, the target function may change over time, an inherent problem of online learning (known as concept drift). In order to handle concept drift regardless of the learning model, we propose new methods to monitor the performance metrics measured during the learning process, to trigger drift signals when a significant variation has been detected. To monitor this performance, we apply some probability inequalities that assume only independent, univariate and bounded random variables to obtain theoretical guarantees for the detection of such distributional changes. Some common restrictions for the online change detection as well as relevant types of change (abrupt and gradual) are considered. Two main approaches are proposed, the first one involves moving averages and is more suitable to detect abrupt changes. The second one follows a widespread intuitive idea to deal with gradual changes using weighted moving averages. The simplicity of the proposed methods, together with the computational efficiency make them very advantageous. We use a Naive Bayes classifier and a Perceptron to evaluate the performance of the methods over synthetic and real data.

Robust user equilibrium model based on cumulative prospect theory under distribution-free travel time

The assumption widely used in the user equilibrium model for stochastic network was that the probability distributions of the travel time were known explicitly by travelers. However, this distribution may be unavailable in reality. By relaxing the restrictive assumption, a robust user equilibrium model based on cumulative prospect theory under distribution-free travel time was presented. In the absence of the cumulative distribution function of the travel time, the exact cumulative prospect value (CPV) for each route cannot be obtained. However, the upper and lower bounds on the CPV can be calculated by probability inequalities. Travelers were assumed to choose the routes with the best worst-case CPVs. The proposed model was formulated as a variational inequality problem and solved via a heuristic solution algorithm. A numerical example was also provided to illustrate the application of the proposed model and the efficiency of the solution algorithm.

Lasso and probabilistic inequalities for multivariate point processes

Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this paper, we consider multivariate counting processes depending on an unknown function parameter to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive $\ell_1$-penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. Nonasymptotic probabilistic results for multivariate Hawkes processes are proven, which allows us to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. Motivated by problems of neuronal activity inference, we finally carry out a simulation study for multivariate Hawkes processes and compare our methodology with the adaptive Lasso procedure proposed by Zou in (J. Amer. Statist. Assoc. 101 (2006) 1418-1429). We observe an excellent behavior of our procedure. We rely on theoretical aspects for the essential question of tuning our methodology. Unlike adaptive Lasso of (J. Amer. Statist. Assoc. 101 (2006) 1418-1429), our tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes, in particular in neuroscience.

On Cauchy–Schwarz’s inequality for Choquet-like integrals without the comonotonicity condition

Cauchy-Schwarz's inequality is one of the most important inequalities in probability, measure theory and analysis. The problem of finding a sharp inequality of Cauchy---Schwarz type for Sugeno integral without the comonotonicity condition based on the multiplication operator has led to a challenging and an interesting subject for researchers. In this paper, we give a Cauchy---Schwarz's inequality without the comonotonicity condition based on pseudo-analysis for two classes of Choquet-like integrals as generalizations of Choquet integral and Sugeno integral. In the first class, pseudo-operations are defined by a continuous strictly increasing function $$g$$g. Another class concerns the Choquet-like integrals based on the operator "$$\sup $$sup" and a pseudo-multiplication $$\otimes $$?. When working on the second class of Choquet-like integrals, our results give a new version of Cauchy---Schwarz's inequality for Sugeno integral without the comonotonicity condition based on the multiplication operator.

Markov's Inequality and Chebyshev's Inequality for Tail Probabilities: A Sharper Image

Markov's inequality gives an upper bound on the probability that a nonnegative random variable takes large values. For example, if the random variable is the lifetime of a person or a machine, Markov's inequality says that the probability that an individual survives more than three times the average lifetime in the population of such individuals cannot exceed one-third. Here we give a simple, intuitive geometric interpretation and derivation of Markov's inequality. These results lead to inequalities sharper than Markov's when information about conditional expectations is available, as in reliability theory, demography, and actuarial mathematics. We use these results to sharpen Chebyshev's tail inequality also.

Geometrically convergent sequences of upper and lower bounds on the Wallis ratio and related expressions

Sequences of algebraic upper and lower bounds on the Wallis ratio ( x + 1)/( x + 1 ) are given with the relative errors that converge to 0 geometrically and uniformly on any interval of the form (x0,1) for x0 > 1 ; moreover, the relative and absolute errors converge to 0 as x ! 1. These conclusions are based on corresponding results for the digamma function := 0 /. Relations with other relevant results are discussed, as well as the corresponding computational aspects. This work was motivated by studies of exact bounds involving the Student probability distribution. AMS 2000 subject classifications: Primary 33B15, 26D07, 26D15, 41A17; secondary 33F05, 65D20, 60E15, 62E15, 62E17. Keywords and phrases: Gamma function, digamma function, Wallis ra- tio, upper bounds, lower bounds, exact bounds, inequalities in approxima- tion, Student's distribution, Student's statistic, probability inequalities.

Probabilistic Inequalities for the Galton--Watson Processes

Probabilistic inequalities are deduced for all values of a breeding coefficient $A$ of a branching process. Previously, only the case A=1 was considered. Separately the cases are studied such that the Cramer condition is fulfilled and there exists only the finite number of moments. Thereafter one of the obtained inequalities is applied for deducing the Rosenthal-type moment inequality for $A\ge 1$.

On the equivalence of the Clauser–Horne and Eberhard inequality based tests

Recently, the results of the first experimental test for entangled photons closing the detection loophole (also referred to as the fair sampling loophole) were published (Vienna, 2013). From the theoretical viewpoint the main distinguishing feature of this long-aspired to experiment was that the Eberhard inequality was used. Almost simultaneously another experiment closing this loophole was performed (Urbana-Champaign, 2013) and it was based on the Clauser–Horne inequality (for probabilities). The aim of this note is to analyze the mathematical and experimental equivalence of tests based on the Eberhard inequality and various forms of the Clauser–Horne inequality. The structure of the mathematical equivalence is nontrivial. In particular, it is necessary to distinguish between algebraic and statistical equivalence. Although the tests based on these inequalities are algebraically equivalent, they need not be equivalent statistically, i.e., theoretically the level of statistical significance can drop under transition from one test to another (at least for finite samples). Nevertheless, the data collected in the Vienna test implies not only a statistically significant violation of the Eberhard inequality, but also of the Clauser–Horne inequality (in the ratio-rate form): for both a violation .

Estimates on compressed neural networks regression

When the neural element number n of neural networks is larger than the sample size m, the overfitting problem arises since there are more parameters than actual data (more variable than constraints). In order to overcome the overfitting problem, we propose to reduce the number of neural elements by using compressed projection A which does not need to satisfy the condition of Restricted Isometric Property (RIP). By applying probability inequalities and approximation properties of the feedforward neural networks (FNNs), we prove that solving the FNNs regression learning algorithm in the compressed domain instead of the original domain reduces the sample error at the price of an increased (but controlled) approximation error, where the covering number theory is used to estimate the excess error, and an upper bound of the excess error is given.

A result on precise asymptotics for largest eigenvalues of β ensembles

The paper focuses on the precise asymptotics of the largest eigenvalues of β-Hermite ensemble and β-Laguerre ensemble. In particular, we obtain a general law on the precise moment convergence rates for a type of weighted series constructed by the first-order conditional moment of the largest eigenvalues of β ensembles. The results are motivated by the complete convergence for partial sums of independent random variables. The proofs depend on the small deviations for largest eigenvalue of β ensembles and non-asymptotic tail probability inequalities of the general β Tracy-Widom law.

Ruin Probability in a Generalized Risk Process under Rates of Interest with Dependent Structures

The aim of this paper is to build recursive and integral equations for ruin probabilities of generalized risk processes u nder rates of interest with homogenous markov chain claims and homogenous markov chain premiums, while the interest rates follow a first- order autoregressive processe. Generalized Lundberg inequalities for ruin probabilities of this processe are derive d by using recursive technique.