Tail Inequalities Research Articles

Network homophily via tail inequalities.

Homophily is the principle whereby "similarity breeds connections." We give a quantitative formulation of this principle within networks. Given a network and a labeled partition of its vertices, the vector indexed by each class of the partition, whose entries are the number of edges of the subgraphs induced by the corresponding classes, is viewed as the observed outcome of the random vector described by picking labeled partitions at random among labeled partitions whose classes have the same cardinalities as the given one. This is the recently introduced random coloring model for network homophily. In this perspective, the value of any homophily score Θ, namely, a nondecreasing real-valued function in the sizes of subgraphs induced by the classes of the partition, evaluated at the observed outcome, can be thought of as the observed value of a random variable. Consequently, according to the score Θ, the input network is homophillic at the significance level α whenever the one-sided tail probability of observing a value of Θ at least as extreme as the observed one is smaller than α. Since, as we show, even approximating α is an NP-hard problem, we resort to classical tails inequality to bound α from above. These upper bounds, obtained by specializing Θ, yield a class of quantifiers of network homophily. Computing the upper bounds requires the knowledge of the covariance matrix of the random vector, which was not previously known within the random coloring model. In this paper we close this gap. Interestingly, the matrix depends on the input partition only through the cardinalities of its classes and depends on the network only through its degrees. Furthermore all the covariances have the same sign, and this sign is a graph invariant. Plugging this structure into the bounds yields a meaningful, easy to compute class of indices for measuring network homophily. As demonstrated in real-world network applications, these indices are effective and reliable, and may lead to discoveries that cannot be captured by the current state of the art.

Some new inequalities for beta distributions

This note provides some new tail inequalities and exponential inequalities of Hoeffding and Bernstein type for beta distributions.

Generalization of Tail Inequalities for Random Variables in the Martingale Theory

We generalize the tail Doob's inequality, concerning two non-negative random variables, arising in the martingale theory, in three directions: on the more general source data, on the random variables belonging to the so-called Grand Lebesgue Spaces, as well as on the multidimensional variables. We also provide several examples. Moreover we show the exactness of the estimates obtained in the particular case of positive random variables having exponential distribution.

A chance-constrained optimization framework for wind farms to manage fleet-level availability in condition based maintenance and operations

Operations and maintenance (O&M) is a key contributor to wind farm expenditures. To increase competitiveness, wind farm operators are increasingly looking into leveraging real-time sensor data from condition monitoring (CM) systems. CM provides significant insights on evolving asset failure risks for wind turbines. To date, these insights have not been fully leveraged in wind farm O&M due to ad-hoc connections to decision-making. Specifically, CM applications in wind farms have been limited to detection of turbines with imminent failure risks that require immediate replacement. In reality, wind farm maintenance requires a careful proactive orchestration of O&M dependencies across turbines along with multiple sources of uncertainty associated with asset availability, operational and market conditions. This paper proposes a unified condition-based maintenance and operations scheduling approach for wind farms that models uncertainties related to turbine availability, wind power output and market price. The proposed formulation explicitly considers the turbine-to-turbine dependencies in operations and maintenance, such as opportunistic maintenance, to identify the O&M decisions that are optimal for multiple wind farms. The problem is formulated as a chance-constrained stochastic programming model to maximize operational revenue while ensuring high levels of turbine availability and generation. To make the chance constraints tractable, two approximations are proposed with a focus on sample average approximation (SAA) and prominent tail inequalities such as Markov’s inequality and Chernoff bound. Our results on a comprehensive set of experiments demonstrate that the proposed approach provides significant improvements in asset availability, market revenue and maintenance costs in large scale wind farms.

Fair resource allocation for demands with sharp lower tail inequalities

We consider a fairness problem in resource allocation where multiple groups demand resources from a common source with the total fixed amount. We show that for many common demand distributions satisfying sharp lower-tail inequalities, a natural allocation that provides resources proportional to each group's average demand performs very well. More specifically, this natural allocation is approximately fair and efficient (i.e., it provides near maximum utilization). We also show that, when a small amount of unfairness is allowed, the Price of Fairness (PoF), in this case, is close to 1.

Matrix Infinitely Divisible Series: Tail Inequalities and Their Applications

In this paper, we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible (i.d.) series, which is a finite sum of fixed matrices weighted by i.d. random variables. We obtain several types of tail inequalities, including Bennett-type and Bernstein-type inequalities. This allows us to further bound the expectation of the spectral norm of a matrix i.d. series. Moreover, by developing a new lower-bound function for $Q(s)=(s+1)\log(s+1)-s$ that appears in the Bennett-type inequality, we derive a tighter tail inequality of the largest eigenvalue of the matrix i.d. series than the Bernstein-type inequality when the matrix dimension is high. The resulting lower-bound function is of independent interest and can improve any Bennett-type concentration inequality that involves the function $Q(s)$. The class of i.d. probability distributions is large and includes Gaussian and Poisson distributions, among many others. Therefore, our results encompass the existing work \cite{tropp2012user} on matrix Gaussian series as a special case. Lastly, we show that the tail inequalities of a matrix i.d. series have applications in several optimization problems including the chance constrained optimization problem and the quadratic optimization problem with orthogonality constraints. In addition, we also use the resulting tail bounds to show that random matrices constructed from i.d. random variables satisfy the restricted isometry property (RIP) when it acts as a measurement matrix in compressed sensing.

Detecting Structural Differences in Tail Dependence of Financial Time Series

ABSTRACTAn accurate assessment of tail inequalities and tail asymmetries of financial returns is key for risk management and portfolio allocation. We propose a new test procedure for detecting the full extent of such structural differences in the dependence of bivariate extreme returns. We decompose the testing problem into piecewise multiple comparisons of Cramér–von Mises distances of tail copulas. In this way, tail regions that cause differences in extreme dependence can be located and consequently be targeted by financial strategies. We derive the asymptotic properties of the test and provide a bootstrap approximation for finite samples. Moreover, we account for the multiplicity of the piecewise tail copula comparisons by adjusting individual p-values according to multiple testing techniques. Monte Carlo simulations demonstrate the test’s superior finite-sample properties for common financial tail risk models, both in the iid and the sequentially dependent case. During the last 90 years in U.S. stock markets, our test detects up to 20% more tail asymmetries than competing tests. This can be attributed to the presence of nonstandard tail dependence structures. We also find evidence for diminishing tail asymmetries during every major financial crisis—except for the 2007–2009 crisis—reflecting a risk-return trade-off for extreme returns. Supplementary materials for this article are available online.

Concentration for Poisson U-statistics: Subgraph counts in random geometric graphs

Concentration bounds for the probabilities P(N≥M+r) and P(N≤M−r) are proved, where M is a median or the expectation of a subgraph count N associated with a random geometric graph built over a Poisson process. The lower tail bounds have a Gaussian decay and the upper tail inequalities satisfy an optimality condition. A remarkable feature is that the underlying Poisson process can have a.s. infinitely many points.The estimates for subgraph counts follow from tail inequalities for more general local Poisson U-statistics. These bounds are proved using recent general concentration results for Poisson U-statistics and techniques involving the convex distance for Poisson processes.

Confidence Corridors for Multivariate Generalized Quantile Regression

We focus on the construction of confidence corridors for multivariate nonparametric generalized quantile regression functions. This construction is based on asymptotic results for the maximal deviation between a suitable nonparametric estimator and the true function of interest, which follow after a series of approximation steps including a Bahadur representation, a new strong approximation theorem, and exponential tail inequalities for Gaussian random fields. As a byproduct we also obtain multivariate confidence corridors for the regression function in the classical mean regression. To deal with the problem of slowly decreasing error in coverage probability of the asymptotic confidence corridors, which results in meager coverage for small sample sizes, a simple bootstrap procedure is designed based on the leading term of the Bahadur representation. The finite-sample properties of both procedures are investigated by means of a simulation study and it is demonstrated that the bootstrap procedure considerably outperforms the asymptotic bands in terms of coverage accuracy. Finally, the bootstrap confidence corridors are used to study the efficacy of the National Supported Work Demonstration, which is a randomized employment enhancement program launched in the 1970s. This article has supplementary materials online.

LSV-Based Tail Inequalities for Sums of Random Matrices.

The techniques of random matrices have played an important role in many machine learning models. In this letter, we present a new method to study the tail inequalities for sums of random matrices. Different from other work (Ahlswede & Winter, 2002 ; Tropp, 2012 ; Hsu, Kakade, & Zhang, 2012 ), our tail results are based on the largest singular value (LSV) and independent of the matrix dimension. Since the LSV operation and the expectation are noncommutative, we introduce a diagonalization method to convert the LSV operation into the trace operation of an infinitely dimensional diagonal matrix. In this way, we obtain another version of Laplace-transform bounds and then achieve the LSV-based tail inequalities for sums of random matrices.

Sharp maximal estimates for BMO martingales

We introduce a method which can be used to study maximal inequalities for martingales of bounded mean oscillation. As an application, we establish sharp $\Phi$-inequalities and tail inequalities for the one-sided maximal function of a BMO martingale. The results can be regarded as BMO counterparts of the classical maximal estimates of Doob.

Some superconcentration inequalities for extrema of stationary Gaussian processes

This note is concerned with concentration inequalities for extrema of stationary Gaussian processes. It provides non-asymptotic tail inequalities which fully reflect the fluctuation rate, and as such improve upon standard Gaussian concentration. The arguments rely on the hypercontractive approach developed by Chatterjee for superconcentration variance bounds. Some statistical illustrations complete the exposition.

Randomized methods for design of uncertain systems: Sample complexity and sequential algorithms

In this paper, we study randomized methods for feedback design of uncertain systems. The first contribution is to derive the sample complexity of various constrained control problems. In particular, we show the key role played by the binomial distribution and related tail inequalities, and compute the sample complexity. This contribution significantly improves the existing results by reducing the number of required samples in the randomized algorithm. These results are then applied to the analysis of worst-case performance and design with robust optimization. The second contribution of the paper is to introduce a general class of sequential algorithms, denoted as Sequential Probabilistic Validation (SPV). In these sequential algorithms, at each iteration, a candidate solution is probabilistically validated, and corrected if necessary, to meet the required specifications. The results we derive provide the sample complexity which guarantees that the solutions obtained with SPV algorithms meet some pre-specified probabilistic accuracy and confidence. The performance of these algorithms is illustrated and compared with other existing methods using a numerical example dealing with robust system identification.

The moment convergence rates for largest eigenvalues of β ensembles

The paper focuses on the largest eigenvalues of the β-Hermite ensemble and the β-Laguerre ensemble. In particular, we obtain the precise moment convergence rates of their largest eigenvalues. The results are motivated by the complete convergence for partial sums of i.i.d. random variables, and the proofs depend on the small deviations for largest eigenvalues of the β ensembles and tail inequalities of the general β Tracy-Widom law.

The use of tail inequalities on the probable computational time of randomized search heuristics

For the purpose of analyzing the time cost of evolutionary algorithms (EAs) or other types of randomized search heuristics (RSHs) with certain requirements on the probability of obtaining a target solution, this paper proposes a new index, called the probable computational time (PCT), which complements expected running time analysis. Using simple tail inequalities, such as Markov’s inequality and Chebyshev’s inequality, we also provide basic properties of PCT, explicitly exhibiting the general relationships between the expected running time and the PCT. To present deeper estimations of the PCT for specific RSHs and problems, we demonstrate a new inequality that is based on the general form of the Chernoff inequality and previous methods such as “fitness-based partitions” and “potential functions”, which have been used to analyze the expected running time of RSHs. The precondition of the new inequality is that the total running time can be described as the sum of a linear combination of some independent geometrically distributed variables and a constant term. The new inequality always provides meaningful upper bounds for the PCT under such circumstances. Some applications of the new inequality for simple EAs, ant colony optimization (ACO) and particle swarm optimization (PSO) algorithms on simple pseudo-Boolean functions are illustrated in this paper.

Tail inequalities for sums of random matrices that depend on the intrinsic dimension

This work provides exponential tail inequalities for sums of random matrices that depend only on intrinsic dimensions rather than explicit matrix dimensions. These tail inequalities are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit matrix dimensions replaced by a trace quantity that can be small even when the explicit dimensions are large or infinite. Some applications to covariance estimation and approximate matrix multiplication are given to illustrate the utility of the new bounds.

Concentration of measure for the number of isolated vertices in the Erdös–Rényi random graph by size bias couplings

Let Y be a nonnegative random variable with mean μ and let Y s , defined on the same space as Y , have the Y -size biased distribution, that is, the distribution characterized by E [ Y f ( Y ) ] = μ E [ f ( Y s ) ] for all functions f for which these expectations exist. The size bias coupling of Y to Y s can be used to obtain the following concentration of measure result when Y counts the number of isolated vertices in an Erdös- Rényi random graph model on n edges with edge probability p . With σ 2 denoting the variance of Y , P ( Y − μ σ ≥ t ) ≤ inf θ ≥ 0 exp ( − θ t + H ( θ ) ) where H ( θ ) = μ 2 σ 2 ∫ 0 θ s γ s d s with γ s = 2 e 2 s ( 1 + p e s 1 − p ) n + ( 1 − p ) − n + 1 . Left tail inequalities may be obtained in a similar fashion. When n p → c for some constant c ∈ ( 0 , ∞ ) as n → ∞ , the bound is of the order at most e − ρ t for some positive constant ρ .

Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates

Let $f=(f_n)_{n\geq 0}$ be a nonnegative submartingale starting from $x$ and let $g=(g_n)_{n\geq 0}$ be a sequence starting from $y$ and satisfying $$|dg_n|\leq |df_n|,\quad |\mathbb{E}(dg_n|\mathcal{F}_{n-1})|\leq \mathbb{E}(df_n|\mathcal{F}_{n-1})$$ for $n\geq 1$. We determine the best universal constant $U(x,y)$ such that $$\mathbb{P}(\sup_ng_n\geq 0)\leq ||f||_1+U(x,y).$$ As an application, we deduce a sharp weak type $(1,1)$ inequality for the one-sided maximal function of $g$ and determine, for any $t\in [0,1]$ and $\beta\in\mathbb{R}$, the number $$ L(x,y,t,\beta)=\inf\{||f||_1: \mathbb{P}(\sup_ng_n\geq \beta)\geq t\}.$$ The estimates above yield analogous statements for stochastic integrals in which the integrator is a nonnegative submartingale. The results extend some earlier work of Burkholder and Choi in the martingale setting.

Moment inequalities for U-statistics

We present moment inequalities for completely degenerate Banach space valued (generalized) U-statistics of arbitrary order. The estimates involve suprema of empirical processes which, in the real-valued case, can be replaced by simpler norms of the kernel matrix (i.e., norms of some multilinear operators associated with the kernel matrix). As a corollary, we derive tail inequalities for U-statistics with bounded kernels and for some multiple stochastic integrals.

VaR as a risk measure for multiperiod static inventory models

This paper explores the possibility to use value-at-risk (VaR) in the context of inventory management. VaR is being used more and more in financial management as a natural measure of the risk taken with a given position. In the framework of inventory management it can work as well. After having built a decision model, where the choice concerns the quantity to be ordered to face a random demand, in order to optimize the expected result (an expected cost to be minimized or an expected profit to be maximized) the model explores the probability distribution of the result, both via analytical methods and via simulation methods. The analytical exploration of this problem has originated a general method to deduce from inequalities for distribution functions, which are based on the expected value, related tail inequalities, which are particularly useful for VaR problems, where only one tail is involved.