Probability Inequalities Research Articles

Exponential inequalities for the number of subgraphs in the Erdös–Rényi random graph

Let Rn be the centered and normalized number of copies of a fixed graph G contained in the Erdös–Rényi graph with n vertices. We obtain upper exponential inequalities for the tail probabilities of Rn, which are uniform in n.

Distributional inequalities for noncommutative martingales

We establish distributional estimates for noncommutative martingales, in the sense of decreasing rearrangements of the spectra of unbounded operators, which generalises the study of distributions of random variables. Our results include distributional versions of the noncommutative Stein, dual Doob, martingale transform and Burkholder-Gundy inequalities. Our proof relies upon new and powerful extrapolation theorems. As an application, we obtain some new martingale inequalities in symmetric quasi-Banach operator spaces and some interesting endpoint estimates. Our main approach demonstrates a method to build the noncommutative and classical probabilistic inequalities in an entirely operator theoretic way.

Colorectal cancer trends in Chile: A Latin-American country with marked socioeconomic inequities.

Colorectal cancer (CRC) is the third most frequent malignant disease in the world. In some countries with established screening programs, its incidence and mortality have decreased, and survival has improved. To obtain reliable data about the epidemiology of CRC in Chile, we analyzed the trends in the last ten years and the influence of observable factors on survival, including a nationwide health program for CRC treatment access (GES program). Publicly available data published by the Ministry of Health and National Institute of Statistics were used. Data were obtained from registries of mortality and hospital discharges, making follow-up of the individuals possible. Crude and age-standardized incidence and mortality rates were calculated, and individual survival was studied by constructing Kaplan-Meier curves. Finally, a Cox statistical model was established to estimate the impact of the observable factors. We found 37,217 newly identified CRC patients between 2008 and 2019 in Chile, corresponding to 103,239 hospital discharges. In the same period, 24,217 people died of CRC. A nearly linear, steady increase in crude incidence, mortality and prevalence was observed. CRC incidence was the lowest in the North of the country, increasing toward the South and reaching a maximum value of 34.6/100,000 inhabitants/year in terms of crude incidence and 20.7/100,000 inhabitants/year in terms of crude mortality in the XII region in 2018. We found that older patients had lower survival rates, as well as men compared to women. Survival was significantly better for patients with private insurance than those under the public insurance system, and the treating hospital also played a significant role in the survival of patients. Patients in the capital region survived longer than those in almost every other part of the country. We found no significant effect on survival associated with the GES program. We found important inequalities in the survival probabilities for CRC patients in Chile. Survival depends mainly on the type of insurance, access to more complex hospitals, and geographical location; all three factors correlated with socioeconomic status of the population. Our work emphasized the need to create specific programs addressing primary causes to decrease the differences in CRC survival.

Pairwise learning problems with regularization networks and Nyström subsampling approach

Pairwise learning usually refers to the learning problem that works with pairs of training samples, such as ranking, similarity and metric learning, and AUC maximization. To overcome the challenge of pairwise learning in the large scale computation, this paper introduces Nyström sampling approach to the coefficient-based regularized pairwise algorithm in the context of kernel networks. Our theorems establish that the obtained Nyström estimator achieves the minimax error over all estimators using the whole data provided that the subsampling level is not too small. We derive the function relation between the subsampling level and regularization parameter that guarantees computation cost reduction and asymptotic behaviors’ optimality simultaneously. The Nyström coefficient-based pairwise learning method does not require the kernel to be symmetric or positive semi-definite, which provides more flexibility and adaptivity in the learning process. We apply the method to the bipartite ranking problem, which improves the state-of-the-art theoretical results in previous works. By developing probability inequalities for U-statistics on Hilbert–Schmidt operators, we provide new mathematical tools for handling pairs of examples involved in pairwise learning.

Improved probability inequalities for Mardia’s coefficient of kurtosis

We provide a probability inequality which, for some classes of multivariate distributions, improves on probability inequalities related to Mardia’s coefficient of kurtosis. Furthermore, we extend our result to the case of a random vector with singular covariance matrix.

Precise Asymptotics for the Uniform Empirical Process and the Uniform Sample Quantile Process

One of the sources of “invariance principle” is that the limit properties of the uniform empirical process coincide with that of a Brownian bridge. The deep discussion of limit theorem of the uniform empirical process gathered wild interest of the researchers. In this paper, the precise convergence rate of the uniform empirical process is considered. As is well-known, when ε tends to 0, the precise asymptotic theorems can be demonstrated by referring to the classical method of Gut and Spǎtaru, by using some nice probability inequalities and so on. However, if ε tends to a positive constant, other powerful methods and tools are needed. The method of strong approximation is used in this paper. The main theorems are proved by using the Brownian bridge B t to approximate the uniform empirical process α n t . The relevant results for the uniform sample quantile process are also presented.

Some matrix inequalities and applications to probabilistic inequalities

Some matrix inequalities and applications to probabilistic inequalities

Jensen's and Cantelli's inequalities with imprecise previsions

We investigate how basic probability inequalities can be extended to an imprecise framework, where (precise) probabilities and expectations are replaced by imprecise probabilities and lower/upper previsions. We focus on inequalities giving information on a single bounded random variable X, considering either convex/concave functions of X (Jensen's inequalities) or one-sided bounds such as (X≥c) or (X≤c) (Markov's and Cantelli's inequalities). As for the consistency of the relevant imprecise uncertainty measures, our analysis considers coherence as well as weaker requirements, notably 2-coherence, which proves to be often sufficient. Jensen-like inequalities are introduced, as well as a generalisation of a recent improvement to Jensen's inequality. Some of their applications are proposed: extensions of Lyapunov's inequality and inferential problems. After discussing upper and lower Markov's inequalities, Cantelli-like inequalities are proven with different degrees of consistency for the related lower/upper previsions. In the case of coherent imprecise previsions, the corresponding Cantelli's inequalities make use of Walley's lower and upper variances, generally ensuring better bounds.

Moving beyond sub-Gaussianity in high-dimensional statistics: applications in covariance estimation and linear regression

Abstract Concentration inequalities form an essential toolkit in the study of high-dimensional statistical methods. Most of the relevant statistics literature in this regard is, however, based on the assumptions of sub-Gaussian or sub-exponential random variables/vectors. In this paper, we first bring together, through a unified exposition, various probabilistic inequalities for sums of independent random variables under much more general exponential type (namely sub-Weibull) tail assumptions. These results extract a part sub-Gaussian tail behavior of the sum in finite samples, matching the asymptotics governed by the central limit theorem, and are compactly represented in terms of a new Orlicz quasi-norm—the Generalized Bernstein–Orlicz norm—that typifies such kind of tail behaviors. We illustrate the usefulness of these inequalities through the analysis of four fundamental problems in high-dimensional statistics. In the first two problems, we study the rate of convergence of the sample covariance matrix in terms of the maximum elementwise norm and the maximum $k$-sub-matrix operator norm that are key quantities of interest in bootstrap procedures and high-dimensional structured covariance matrix estimation, as well as in high-dimensional and post-selection inference. The third example concerns the restricted eigenvalue condition, required in high-dimensional linear regression, which we verify for all sub-Weibull random vectors through a unified analysis, and also prove a more general result related to restricted strong convexity in the process. In the final example, we consider the Lasso estimator for linear regression and establish its rate of convergence to be generally $\sqrt{k\log p/n}$, for $k$-sparse signals, under much weaker than usual tail assumptions (on the errors as well as the covariates), while also allowing for misspecified models and both fixed and random design. To our knowledge, these are the first such results for Lasso obtained in this generality. The common feature in all our results over all the examples is that the convergence rates under most exponential tails match the usual (optimal) ones obtained under sub-Gaussian assumptions. Finally, we also establish some complementary results on analogous tail bounds for the suprema of empirical processes indexed by sub-Weibull variables. All our results are finite samples.

On the linearly extended negative quadrant dependent random variables and its inequalities

In this article, the structure of linearly extended negative quadrant dependent (LENQD) random variables is given, and its basic properties are investigated. In addition, the characteristic function inequality, Bernstein-type inequality, some probability inequalities, and the Rosenthal-type inequality are established for LENQD random variables.

Maximal Probability Inequalities in Vector Lattices

Maximal Probability Inequalities in Vector Lattices

Relaxation schemes for the joint linear chance constraint based on probability inequalities

<p style='text-indent:20px;'>This paper is concerned with the joint chance constraint for a system of linear inequalities. We discuss computationally tractble relaxations of this constraint based on various probability inequalities, including Chebyshev inequality, Petrov exponential inequalities, and others. Under the linear decision rule and additional assumptions about first and second order moments of the random vector, we establish several upper bounds for a single chance constraint. This approach is then extended to handle the joint linear constraint. It is shown that the relaxed constraints are second-order cone representable. Numerical test results are presented and the problem of how to choose proper probability inequalities is discussed.</p>

A Fundamental Study for Essentially Improving Performance of Hoeffding Probability Inequality

A Fundamental Study for Essentially Improving Performance of Hoeffding Probability Inequality

Bounds for the expected supremum of some non-stationary Gaussian processes

The aim of this article is to obtain closed-form lower and upper bounds for the expectation of the supremum of some non-stationary Gaussian processes. This supremum is an important quantity in applications such as finance and queueing systems. The covariance function structure, decomposition of the processes and probability inequalities are applied for some examples of non-stationary Gaussian processes, including sub-fractional, bifractional and multifractional Brownian motions. Malliavin calculus operators are also applied in some cases to find bounds for the density and expectation of the supremum of these processes.

Robust Hybrid Precoding Design for Securing Millimeter-Wave IoT Networks Under Secrecy Outage Constraint

Hybrid precoding architecture, as a cost-effective approach for millimeter-wave (mmWave) communications, can achieve an excellent tradeoff between spectrum efficiency and hardware implementation complexity. However, the design of a robust hybrid precoding for improving the physical layer security (PLS), which is insensitive to the uncertainty of eavesdropper's channel state information (CSI), has not been well studied. This article for the first time designs a probabilistically robust hybrid precoding scheme for securing broadcast communications in Internet of Things (IoT) with eavesdropper's imperfect CSI. Specifically, considering the Gaussian CSI error model, we maximize the minimum secrecy rate of multiple IoT devices (IoDs) by jointly designing analog and digital precoders under the constraints in terms of secrecy outage probability and per IoD's information rate. The optimization problem is challenging due to the coupling of the analog and digital precoders, and the secrecy outage constraint. To handle these challenges, we first employ a conservative probability inequality to transform the secrecy outage probability constraint into a deterministic one. Then, by employing the penalty dual decomposition (PDD) method, we develop a novel iterative algorithm to convert the resultant nonconvex problem into a sequence of convex problems, which can guarantee the convergence to its Karush-Kuhn-Tucker (KKT) solution. Simulation results show that the proposed algorithm can achieve significant secrecy performance gains compared with the benchmark algorithm.

Zero-free neighborhoods around the unit circle for Kac polynomials

In this paper, we study how the roots of the Kac polynomials W_n(z) = sum _{k=0}^{n-1} xi _k z^k concentrate around the unit circle when the coefficients of W_n are independent and identically distributed nondegenerate real random variables. It is well known that the roots of a Kac polynomial concentrate around the unit circle as nrightarrow infty if and only if {mathbb {E}}[log ( 1+ |xi _0|)]<infty . Under the condition {mathbb {E}}[xi ^2_0]<infty , we show that there exists an annulus of width {text {O}}(n^{-2}(log n)^{-3}) around the unit circle which is free of roots with probability 1-{text {O}}({(log n)^{-{1}/{2}}}). The proof relies on small ball probability inequalities and the least common denominator used in [17].

Normalized fractional inequalities for continuous random variables

We introduce new normalized fractional integral concepts on random variables. We also introduce the fractional coupled random variables. The Jensen integral inequality is generalized. Also, some fractional bounds estimating the expectation and variance are delivered. At the end, some other results related to the coupled random variables are proved. For this work, some results of the papers [On some statistical and probability inequalities. Journal of Inequalities and Special Functions, 2016] and [Some inequalities for the expectation and variance of a random variable whose PDFs are absolutely continuous using a pre-Chebychev inequality. Tamkang Journal of Mathematics, 2001] are deduced as some special cases.

NEW EXPONENTIAL PROBABILITY INEQUALITY AND COMPLETE CONVERGENCE FOR F-LNQD RANDOM VARIABLES SEQUENCE WITH APPLICATION TO AR(1)MODEL GENERATED BY F-LNQD ERRORS

The exponential probability inequalities have been important tools in probability and statistics. In this paper, we prove a new tail probability inequality for the distri-butions of sums of conditionally linearly negative quadrant dependent (F-LNQD , in short) random variables, and obtain a result dealing with conditionally complete con-vergence of first-order autoregressive processes with identically distributed (F-LNQD) innovations.

Probability inequalities for multiplicative sequences of random variables

We extend some sharp inequalities for martingale-differences to general multiplicative systems of random variables. The key ingredient in the proofs is a technique reducing the general case to the case of Rademacher random variables without change of the constants in inequalities.

Error analysis of the kernel regularized regression based on refined convex losses and RKBSs

In this paper, we bound the errors of kernel regularized regressions associating with [Formula: see text]-uniformly convex two-sided RKBSs and differentiable [Formula: see text] uniformly smooth losses. In particular, we give learning rates for the learning algorithm with loss [Formula: see text]. Also, we show a probability inequality and with which provide the error bounds for kernel regularized regression with loss [Formula: see text] The discussions are comprehensive applications of the uniformly smooth function theory, the uniformly convex function theory and uniformly convex space theory.