Markov Inequality Research Articles

Reverse Markov Inequality on the Unit Interval for Polynomials Whose Zeros Lie in the Upper Unit Half-Disk

We prove that there is an absolute constant A > 0 such that $$\begin{array}{l} \begin{array}{*{20}{c}} {\max } \\ { - 1 \le x \le 1} \\ \end{array}|P'(x)| \ge A\sqrt {n \cdot } \begin{array}{*{20}{c}} {\max } \\ { - 1 \le x \le 1} \\ \end{array}\,|P(x)| \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{array}$$ for an arbitrary algebraic polynomial P of degree n whose zeros lie in the half-disk $$\left\{ {z:|z| \le 1,{\mathop{\rm Im}\nolimits} z \ge 0} \right\}$$.

Halving the Bounds for the Markov, Chebyshev, and Chernoff Inequalities Using Smoothing

The Markov, Chebyshev, and Chernoff inequalities are some of the most widely used methods for bounding the tail probabilities of random variables. In all three cases, the bounds are tight in the sense that there exists easy examples where the inequalities become equalities. Here we will show, through a simple smoothing using auxiliary randomness, that each of the three bounds can be cut in half. In many common cases, the halving can be achieved without the need for the auxiliary randomness.

The stability analysis for uncertain heat equations based on p-th moment

Stability in p-th moment plays a vital role in uncertain heat equation (UHE). However, little is known about the definition and properties of stability in p-th moment for UHE. This paper fills this gap and advances the concept of stability in p-th moment for UHE. Based on Markov inequality, this study shows the relationship between stability in p-th moment and stability in measure, that is if an UHE is stable in p-th moment, then it is stable in measure, but not vice versa. Our analysis further reveals that if the coefficients of UHE satisfy strong Lipschitz condition, and meanwhile, the Lipschitz coefficients meet some integral constraints, then the UHE is stable in p-th moment. This study provides a strong theoretically foundation for understanding the stability in p-th moment of UHE.

Improvements of the Markov and Chebyshev inequalities using the partial expectation

The multivariate Markov and multiple Chebyshev inequalities are improved using the partial expectation. The conditions of the strict inequalities between the improved and unimproved bounds of probabilities are given. The known upper bounds of the tail probabilities by the partial moment are derived as special cases of the Markov inequality, whose improved bounds and extensions to multivariate cases using the semicovariances and partial product moment are presented.

Theorems of complete convergence and complete integral convergence for END random variables under sub-linear expectations

The goal of this paper is to build complete convergence and complete integral convergence for END sequences of random variables under sub-linear expectation space. By using the Markov inequality, we extend some complete convergence and complete integral convergence theorems for END sequences of random variables when we have a sub-linear expectation space, and we provide a way to learn this subject.

On the best constants in Markov inequalities for the Laplace operator on polynomials with the Hermite and Legendre norm

We consider the Laplace operator on the finite-dimensional linear space of algebraic polynomials in N variables such that each variable occurs at most with the power n. This space of polynomials is equipped with the Hermite norm or the Legendre norm. We derive an asymptotic formula for the norm of the Laplace operator on this space in the Hermite case and asymptotic upper and lower bounds in the Legendre case as n goes to infinity.

EMBB-URLLC Resource Slicing: A Risk-Sensitive Approach

Ultra Reliable Low Latency Communication (URLLC) is a 5G New Radio (NR) application that requires strict reliability and latency. URLLC traffic is usually scheduled on top of the ongoing enhanced Mobile Broadband (eMBB) transmissions (i.e., puncturing the current eMBB transmission) and cannot be queued due to its hard latency requirements. In this letter, we propose a risk-sensitive based formulation to allocate resources to the incoming URLLC traffic while minimizing the risk of the eMBB transmission (i.e., protecting the eMBB users with low data rate) and ensuring URLLC reliability. Specifically, the Conditional Value at Risk (CVaR) is introduced as a risk measure for eMBB transmission. Moreover, the reliability constraint of URLLC is formulated as a chance constraint and relaxed based on Markov's inequality. We decompose the formulated problem into two subproblems in order to transform it into a convex form and then alternatively solve them until convergence. Simulation results show that the proposed approach allocates resources to the incoming URLLC traffic efficiently while satisfying the reliability of both eMBB and URLLC.

Equivalence of the global and local Markov inequalities in the complex plane

We prove that a compact subset of the complex plane satisfies the Global Markov Inequality if and only if it admits an extension property and a Kolmogorov type inequality in Jackson norms, generalizing a result established by Bos and Milman in the real case. We also show that the Global Markov Inequality is equivalent to the Local Markov Property for all compact subsets of the complex plane admitting the Jackson Property. The latter is a generalization of Jackson's approximation inequality, closely connected with the boundary behavior of the Green's function. Finally, we construct an example showing that there is no equivalence in general.

The multivariate Markov and multiple Chebyshev inequalities

A sharp probability inequality named the multivariate Markov inequality is derived for the intersection of the survival functions for non-negative random variables as an extension of the Markov inequality for a single variable. The corresponding result in Chebyshev’s inequality is also obtained as a special case of the multivariate Markov inequality, which is called the multiple Chebyshev inequality to distinguish from the multivariate Chebyshev inequality for a quadratic form of standardized uncorrelated variables. Further, the results are extended to the inequalities for the union of the survival functions and those with lower bounds.

Optimizing the confidence bound of count-min sketches to estimate the streaming big data query results more precisely

A count-min sketch is a probabilistic data structure, which serves as a frequency table of events to process a stream of big data. It uses hash functions to map events to frequencies. Querying a count-min sketch returns the targeted event along with an estimated frequency, which is not less than the actual frequency. The estimated error, i.e., the difference between the estimated frequency and the actual, can be measured by a pre-defined confidence bound. However, the bound originally defined is too loose. The reason is that the Markov inequality used to derive the bound does not perform well. In this paper, based on binomial distribution and central limit theorem, we define a tighter bound. We indicate that the reliability of the bound is related to the deviation of data, which can be measured by the data’s coefficient of standard deviation. Our extensive experiments well support the effectiveness and efficiency of the new bound.

Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies

We construct norming meshes for polynomial optimization by the classical Markov inequality on general convex bodies in $${\mathbb {R}}^d$$ , and by a tangential Markov inequality via an estimate of the Dubiner distance on smooth convex bodies. These allow to compute a $$(1-\varepsilon )$$ -approximation to the minimum of any polynomial of degree not exceeding n by $${\mathcal {O}}\left( (n/\sqrt{\varepsilon })^{\alpha d}\right) $$ samples, with $$\alpha =2$$ in the general case, and $$\alpha =1$$ in the smooth case. Such constructions are based on three cornerstones of convex geometry, Bieberbach volume inequality and Leichtweiss inequality on the affine breadth eccentricity, and the Rolling Ball Theorem, respectively.

Proof-without-words: Markov's inequality

Proof-without-words: Markov's inequality

Almost surely attractive sets of discrete‐time Markov jump systems with stochastic disturbances via impulsive control

In this study, the problem on the globally almost surely attractive sets is addressed for a class of discrete-time Markov jump systems with stochastic disturbances via impulsive control. Based on the Lyapunov function methods and the Markov's inequality, the authors derive some sufficient conditions guaranteeing the existence of the global almost surely attractive sets of the considered systems. Meanwhile, the estimation of the attractive sets is also given out. It is shown that the unbounded discrete-time Markov jump stochastic system without attractive sets can turn into the bounded one with attractive sets via proper impulsive control strategies. An example is also presented to illustrate the main results.

Workforce Assignment in Assembly Line Considering Uncertain Demand

In this work, we study a workforce assignment problem in the assembly line with several stations and executing some parts of different types. The objective is to minimize the number of workers over all the production line. Due to the complex business environment, changes in demand have great influence on production scheduling. Therefore, we consider uncertain demand in our investigated problem. To the best of our knowledge, we are the first to consider stochastic demand in assembly line problem with workers assignment. The main contributions include: 1) an ambiguity set is applied to portray the demand uncertainty, 2) a joint chance-constrained programming model is proposed, and 3) we choose two distribution-free approaches, namely Approximations based on Markov Inequality (AMI) and Mixed Integer Second-Order Conic Program (MI-SOCP), to approximate the chance constraints of the model. Finally, computational experiments are conducted to compare performances of the two methods.

A Bernstein type inequality for the Askey–Wilson operator

We establish a Riesz-type interpolation formula on the interval [−1,1] for the Askey–Wilson operator. As consequences, sharp Bernstein inequality and Markov inequality are obtained when differentiation is replaced by the Askey–Wilson operator. Moreover, an inverse approximation theorem is proved using a Bernstein type inequality in L2-space. We conclude our paper with an overconvergence result which is applied to characterize all q-differentiable functions of Brown and Ismail.

Sharp constants in weighted $L^2$-Markov inequalities

Sharp constants in weighted $L^2$-Markov inequalities

Stochastic adaptive tracking for a rehabilitative training walker with control constraints considering the omniwheel touchdown characteristic

ABSTRACTThis article investigates the improvement of trajectory tracking on an omnidirectional rehabilitative training walker (ORTW) with random shifts in the centre of gravity, control constraints, and touchdown characteristics of omniwheels. Analysis of the touchdown characteristic improved the accuracy of the dynamic ORTW model. The control constraints guarantee the safety of the omniwheels and the ORTW training process. After describing the random centre-of-gravity shifts using Wiener process, an improved stochastic dynamic model was constructed. Under trajectory tracking control with adaptive technology, the ORTW followed the trajectory designed by the rehabilitation therapist. The exponentially practical stability in mean absolute of the tracking-error system was derived from stochastic theory and Markov's inequality. The effectiveness of the proposed methods was confirmed in simulations.

Optimal factors in Vladimir Markov’s inequality in L2 Norm

In this paper we discuss a problem of computation of constants in Vladimir Markov's type inequality in L2 norm on the interval [-1, 1].

Some Inequalities Combining Rough and Random Information.

Rough random theory, generally applied to statistics, decision-making, and so on, is an extension of rough set theory and probability theory, in which a rough random variable is described as a random variable taking “rough variable” values. In order to extend and enrich the research area of rough random theory, in this paper, the well-known probabilistic inequalities (Markov inequality, Chebyshev inequality, Holder’s inequality, Minkowski inequality and Jensen’s inequality) are proven for rough random variables, which gives a firm theoretical support to the further development of rough random theory. Besides, considering that the critical values always act as a vital tool in engineering, science and other application fields, some significant properties of the critical values of rough random variables involving the continuity and the monotonicity are investigated deeply to provide a novel analytical approach for dealing with the rough random optimization problems.

Regularization under diffusion and anticoncentration of the information content

Under the Ornstein-Uhlenbeck semigroup $\{U_t\}$, any non-negative measurable $f : \mathbb R^n \to \mathbb R_+$ exhibits a uniform tail bound better than that implied by Markov's inequality and conservation of mass: For every $\alpha \geq e^3$, and $t > 0$, \[ \gamma_n\left(\left\{x \in \mathbb R^n : U_t f(x) > \alpha \int f\,d\gamma_n\right\}\right) \leq C(t) \frac{1}{\alpha} \sqrt{\frac{\log \log \alpha}{\log \alpha}}\] where $\gamma_n$ is the $n$-dimensional Gaussian measure and $C(t)$ is a constant depending only on $t$. This confirms positively the Gaussian limiting case of Talagrand's convolution conjecture (1989). This is shown to follow from a more general phenomenon. Suppose that $f : \mathbb{R}^n \to \mathbb{R}_+$ is {\em semi-log-convex} in the sense that for some $\beta > 0$, for all $x \in \mathbb{R}^n$, the eigenvalues of $\nabla^2 \log f(x)$ are at least $-\beta$. Then $f$ satisfies a tail bound asymptotically better than that implied by Markov's inequality.