Probability Inequalities Research Articles

Tail probability bounds for acceptable random variables of partial sums and their application in linear models

Exponential probability inequalities play an important role in many parts of probability and statistics, offering a fundamental tool for tail probability bounding and modeling stochastic processes. The present work proposes a new exponential inequality for acceptable random variables. This class covers dependencies that are not embraced by the standard independence assumptions. Our results extend the theoretical perspective by placing sharper bounds on the above-mentioned variables, which tends to increase the accuracy of probabilistic modeling. Moreover, we study complete convergence in first-order autoregressive processes with acceptable innovations. The complete convergence framework gives further insights into the long-term behavior of the wide class of processes commonly used in time series analysis, econometrics, and other areas of applied research. In turn, our results on the influence of acceptable innovations on the properties of complete convergence establish bridges between theoretical probability inequalities and statistical applications. In addition, these contributions provide new tools in investigating the stability and predictability of the autoregressive models. These considerations consequently set up significant perspectives and enrich a developing body of research related to stochastic processes by reinforcing the theoretical framework as well as the methodological apparatus of probability theory.

ON IMPROVED GAUSSIAN CORRELATION INEQUALITIES FOR SYMMETRICAL -RECTANGLES EXTENDED TO CERTAIN MULTIVARIATE GAMMA DISTRIBUTIONS AND SOME FURTHER PROBABILITY INEQUALITIES

The Gaussian correlation inequality (GCI) for symmetrical n-rectangles is improved if the absolute components have a joint cumulative distribution function (cdf), which is MTP2 (multivariate totally positive of order 2). Inequalities of the given type hold at least for all MTP2-cdfs on or with everywhere positive smooth densities. In particular, at least some infinitely divisible multivariate chi-square distributions (gamma distributions in the sense of Krishnamoorthy and Parthasarathy) with any positive real “degree of freedom” are shown to be MTP2. Moreover, further numerically calculable probability inequalities for a broader class of multivariate gamma distributions are derived. A different improvement for inequalities of the GCI-type, and of a similar type with three instead of two groups of components with more special correlation structures is also obtained. The main idea behind these inequalities is to find for a given correlation matrix with positive correlations, a further correlation matrix with smaller correlations whose inverse is an M-matrix and where the corresponding multivariate gamma distribution function is numerically available.

Tail variance and confidence of using tail conditional expectation: analytical representation, capital adequacy, and asymptotics

Abstract In this paper, we explore the applications of Tail Variance (TV) as a measure of tail riskiness and the confidence level of using Tail Conditional Expectation (TCE)-based risk capital. While TCE measures the expected loss of a risk that exceeds a certain threshold, TV measures the variability of risk along its tails. We first derive analytical formulas of TV and TCE for a large variety of probability distributions. These formulas are useful instruments for relevant research works on tail risk measures. We then propose a distribution-free approach utilizing TV to estimate the lower bounds of the confidence level of using TCE-based risk capital. In doing so, we introduce sharpened conditional probability inequalities, which halve the bounds of conventional Markov and Cantelli inequalities. Such an approach is easy to implement. We further investigate the characterization of tail risks by TV through an exploration of TV’s asymptotics. A distribution-free limit formula is derived for the asymptotics of TV. To further investigate the asymptotic properties, we consider two broad distribution families defined on tails, namely, the polynomial-tailed distributions and the exponential-tailed distributions. The two distribution families are found to exhibit an asymptotic equivalence between TV and the reciprocal square of the hazard rate. We also establish asymptotic relationships between TCE and VaR for the two families. Our asymptotic analysis contributes to the existing research by unifying the asymptotic expressions and the convergence rate of TV for Student-t distributions, exponential distributions, and normal distributions, which complements the discussion on the convergence rate of univariate cases in [28]. To show the usefulness of our results, we present two case studies based on real data from the industry. We first show how to use conditional inequalities to assess the confidence of using TCE-based risk capital for different types of insurance businesses. Then, for financial data, we provide alternative evidence for the relationship between the data frequency and the tail categorization by the asymptotics of TV.

Probability inequalities for strongly left-invariant metric semigroups/monoids, including all lie groups

Probability inequalities for strongly left-invariant metric semigroups/monoids, including all lie groups

Study on applicable coverage extension of theory-based generalization errors bounds to the variants of RVFL network and ELM

This study investigates the coverage extendibility of generalized prediction error estimation for the random vector functional link (RVFL) network and generalized extreme learning machine (GELM). GELM is an ELM applying the limiting behavior of Moore–Penrose generalized inverse (M–P GI) to the output matrix of the last hidden layer of ELM. The RVFL network, the ELM, and their variants are currently employed in various areas due to their fast learning and are expected to be adopted in more applications. Additionally, obtaining prediction error bounds quickly for these networks could greatly increase their utilities. Therefore, exploring the error bounds of the variants is crucial. Recently, the upper and lower bounds of generalized prediction error and their associated tail probabilities for the RVFL network and GELM were analyzed. This study aims to extend the coverage of the existing error bounds to variants of RVFL and ELM. To this end, we investigate each solution of the objective functions for the existing RVFL and ELM variants, compare it with the solution obtained by using the limiting behavior of M–P GI, and then discuss the applicability of the probability inequalities for the prediction error to the variant. Specifically, we conceptualize a restricted solution vector and prove that the restricted solution of a variant of RVFL is identical to that of GELM, with an appropriately selected regularization parameter. The mechanism used in the proof can be applied to other variants, which have different objective functions but have similar forms of solution. Experiments are performed with the CAIDA (Center for Applied Internet Data Analysis) dataset. Numerical results verify the analysis, showing that the generalization prediction error bounds and their associated tail probabilities for ELM (RVFL) variants in applicability coverage are almost identical to those of the GELM (RVFL). In addition, performance-based network similarity is introduced and computed with the obtained results. Based on this analysis, we can conclude that the probability inequalities for prediction error derived can be directly applied to some variants of the RVFL network and ELM, such as robust regularized RVFL and ridge regression ELM. However, it cannot apply directly to the networks that use information obtained during the learning process in its learning, such as RVFL+ and its variants. The significance of this study is that it enables us to figure out the relationship between the size-restricted final weight vector of RVFL and the final weight vectors of ELM, the effect of the relationship on generalized prediction error estimation, and the performance-based similarity between networks. This study provides a theoretical basis for the generalized prediction error's upper and lower bounds and their associated tail probabilities for ELM (RVFL) variants.

Thermodynamics of Computations with Absolute Irreversibility, Unidirectional Transitions, and Stochastic Computation Times

Developing a thermodynamic theory of computation is a challenging task at the interface of nonequilibrium thermodynamics and computer science. In particular, this task requires dealing with difficulties such as stochastic halting times, unidirectional (possibly deterministic) transitions, and restricted initial conditions, features common in real-world computers. Here, we present a framework which tackles all such difficulties by extending the martingale theory of nonequilibrium thermodynamics to generic nonstationary Markovian processes, including those with broken detailed balance and/or absolute irreversibility. We derive several universal fluctuation relations and second-law-like inequalities that provide both lower and upper bounds on the intrinsic dissipation (mismatch cost) associated with any periodic process—in particular, the periodic processes underlying all current digital computation. Crucially, these bounds apply even if the process has stochastic stopping times, as it does in many computational machines. We illustrate our results with exhaustive numerical simulations of deterministic finite automata processing bit strings, one of the fundamental models of computation from theoretical computer science. We also provide universal equalities and inequalities for the acceptance probability of words of a given length by a deterministic finite automaton in terms of thermodynamic quantities, and outline connections between computer science and stochastic resetting. Our results, while motivated from the computational context, are applicable far more broadly. Published by the American Physical Society 2024

Chebyshev–Jensen-Type Inequalities Involving χ-Products and Their Applications in Probability Theory

By means of the functional analysis theory, reorder method, mathematical induction and the dimension reduction method, the Chebyshev-Jensen-type inequalities involving the χ-products ⟨·⟩χ and [·]χ are established, and we proved that our main results are the generalizations of the classical Chebyshev inequalities. As applications in probability theory, the discrete with continuous probability inequalities are obtained.

The law of large numbers for large stable matchings

In many empirical studies of a large two-sided matching market (such as in a college admissions problem), the researcher performs statistical inference under the assumption that they observe a random sample from a large matching market. In this paper, we consider a setting in which the researcher observes either all or a nontrivial fraction of outcomes from a stable matching. We establish a concentration inequality for empirical matching probabilities assuming strong correlation among the colleges’ preferences while allowing students’ preferences to be fully heterogeneous. Our concentration inequality yields laws of large numbers for the empirical matching probabilities and other statistics commonly used in empirical analyses of a large matching market. To illustrate the usefulness of our concentration inequality, we prove consistency for estimators of conditional matching probabilities and measures of positive assortative matching.

On estimation of values of jumps for discrete distribution functions

We obtain new inequalities for values of jumps for discrete distribution functions. Values of jumps are bounded by linear combinations of a finite number of moments of the distributions. Obtained inequalities can be used for estimation of ranges for values of improbable jumps when frequencies are zero and are not interesting as estimators. We discuss relationships of derived inequalities with inequalities for probabilities of unions of events and the Caushy-Bunyakovski and Holder inequalities as well.

Probability Inequalities for Weighted Branching Processes in Random Environments

Probability Inequalities for Weighted Branching Processes in Random Environments

Ultra-quantum coherent states in a single finite quantum system

A set of n coherent states is introduced in a quantum system with d-dimensional Hilbert space H(d). It is shown that they resolve the identity, and also have a discrete isotropy property. A finite cyclic group acts on the set of these coherent states, and partitions it into orbits. A n-tuple representation of arbitrary states in H(d), analogous to the Bargmann representation, is defined. There are two other important properties of these coherent states which make them ‘ultra-quantum’. The first property is related to the Grothendieck formalism which studies the ‘edge’ of the Hilbert space and quantum formalisms. Roughly speaking the Grothendieck theorem considers a ‘classical’ quadratic form that uses complex numbers in the unit disc, and a ‘quantum’ quadratic form that uses vectors in the unit ball of the Hilbert space. It shows that if , the corresponding might take values greater than 1, up to the complex Grothendieck constant . related to these coherent states is shown to take values in the ‘Grothendieck region’ , which is classically forbidden in the sense that does not take values in it. The second property complements this, showing that these coherent states violate logical Bell-like inequalities (which for a single quantum system are quantum versions of the Frechet probabilistic inequalities). In this sense also, our coherent states are deep into the quantum region.

Exponential Inequalities for the Tail Probabilities of the Number of Cycles in Generalized Random Graphs

Let R_n be the centered and normalized number of cycles offixed length contained in a generalized random graph with n vertices. We obtain a Höffding-typeexponential inequality for the tail probability of R_n.

Integrated Transmission Network Planning by Considering Wind Power’s Uncertainty and Disasters

The penetration of wind turbines and other power sources with strong uncertainty into the grid has increased in recent years. It has brought significant technical challenges to power systems’ operation. The volatility and intermittency of wind power increase the risk of insufficient transmission capacity of the lines. Therefore, the traditional deterministic planning methods for transmission grids are no longer fully applicable. On the other hand, the frequent disasters in recent years have posed a great threat to the power system, especially for the transmission grid. This requires the design of transmission lines with high design standards, such as skeleton networks, to withstand disasters. With the aim to address these problems, a bi-level integrated network planning model for the transmission grid is developed by considering wind power’s uncertainty and load guarantee under disasters. Chance constraints are used in the model to characterize wind power’s uncertainty, and a skeleton network is adopted to cope with disasters. Moreover, based on a convex relaxation method, the chance constraints are converted into the probabilistic inequalities to be solved. The proposed method is simulated in the IEEE 118 bus system, and the obtained network planning scheme is further analyzed in the scenario tests. And the result of the tests proves the validity and reasonableness of the proposed method.

Socioeconomic inequalities in cervical precancer screening among women in Ethiopia, Malawi, Rwanda, Tanzania, Zambia and Zimbabwe: analysis of Population-Based HIV Impact Assessment surveys

ObjectivesWe examined age, residence, education and wealth inequalities and their combinations on cervical precancer screening probabilities for women. We hypothesised that inequalities in screening favoured women who were older, lived...

Paradoxes of MaxEnt markedness

Over the past two decades, theoretical linguistics has taken a probabilistic turn. Maximum entropy (ME) has been endorsed as a model of probabilistic phonology because of its classical guarantees for grammatical inference. Yet, little is known about the basic organizing principles of ME phonology beyond circumstantial evidence of ME’s ability to fit specific patterns of empirical frequencies. The study of ME typologies is difficult because they consist of infinitely many grammars that cannot be exhaustively listed and directly inspected. Uniform Probability Inequalities (Anttila and Magri 2018) are a new tool that solves the problem: they characterize cases where one phonological mapping has a probability smaller than another mapping and this probability inequality holds uniformly for every grammar in the typology. In other words, uniform probability inequalities are universals of probabilistic grammars. We present a new generalization about ME uniform probability inequalities and argue that they are phonologically paradoxical and prune the set of ME universals down to almost nothing. This suggests that ME is not a suitable model of phonology.

Optimizing Traffic Light Green Duration under Stochastic Considerations

An optimization model for traffic light control in an urban network of intersections is derived. The model is based on store-and-forward analytic relations, which account for the length of the queue of waiting vehicles in front of the traffic light intersection. The model is complicated with probabilistic relations that formalize the requirements for maintaining short queues of vehicles. Probabilistic inequalities apply to each intersection of the city network. Approximations of probability inequalities are given in the article. Quadratic deterministic inequalities, which are part of the set of the traffic flow control optimization problem, are derived. Numerical simulations are performed, applying mean estimated data for real traffic in an urban area of Sofia. The model predictive approach is applied to traffic light optimization and control. Empirical results give advantages of the obtained model compared to the classical store-and-forward optimization model for the total number of vehicles waiting in the considered urban network.

Probability inequalities for direct and inverse dynamical outputs in driven fluctuating systems.

When a fluctuating system is subjected to a time-dependent drive or nonconservative forces, the direct-inverse symmetry of the dynamics can be broken so inducing an average bias. Here we start from the fluctuation theorem, a cornerstone of stochastic thermodynamics, for inspecting the unbalancing between direct and inverse dynamical outputs, here called "events," in a bidirectional forward-backward setup. The occurrence of an event might correspond to the realization of a quantitative output, or to the realization of a sequence of acts that compose a complex "narrative." The focus is on mutual bounds between the probabilities of occurrence of direct and inverse events in the forward and backward mode. The inspection is made for systems in contact with a thermal bath, and by assuming Markov dynamics on the uncontrolled degrees of freedom. The approach comprises both the case of systems under a time-dependent drive and time-independent external forces. The general formulation is then used to derive (or re-derive) specialized results valid for finite-time processes, and for systems taken into steady conditions (either periodic steady states or steady states) starting from equilibrium. Among the results, we find already known forms of "generalized" thermodynamic uncertainty relations, and derive useful constraints concerning the work distribution function for systems in steady conditions.

Some new hardy inequalities in probability

Some new hardy inequalities in probability

Improvement of Performance of Hoeffding Probability Inequality

Improvement of Performance of Hoeffding Probability Inequality

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.