P. Major
We prove a multivariate version of Hoeffding's inequality about the distribution of homogeneous polynomials of Rademacher functions. The proof is based on such an estimate about the moments of homogeneous polynomials of Rademacher functions which can be considered as an improvement of Borell's inequality in a most important special case.
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This note makes the obvious observation that Hoeffding's original proof of his inequality remains valid in the game-theoretic framework. All details are spelled out for the convenience of future reference.
We give an extension of Hoeffding's inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev.
In this article, we establish Hoeffding's inequality for bounded Lipschitz functions of a class of not necessarily irreducible Markov models. The result complements the existing literature on this topic where Hoeffding's inequality for bounded measurable functions of a class of irreducible Markov models has been considered. Our approach is based on the assumption of uniform ergodicity of the underlying Markov model in $L^1$-Wasserstein space.
In this note, we present a version of Hoeffding's inequality in a continuous-time setting, where the data stream comes from a uniformly ergodic diffusion process. Similar to the well-studied case of Hoeffding's inequality for discrete-time uniformly ergodic Markov chain, the proof relies on techniques ranging from martingale theory to classical Hoeffding's lemma as well as the notion of deviation kernel of diffusion process. We present two examples to illustrate our results. In the first example we consider large deviation probability on the occupation time of the Jacobi diffusion, a popular p…
We show that the Bernstein-Hoeffding method can be employed to a larger class of generalized moments. This class includes the exponential moments whose properties play a key role in the proof of a well-known inequality of Wassily Hoeffding, for sums of independent and bounded random variables whose mean is assumed to be known. As a result we can generalise and improve upon this inequality. We show that Hoeffding's bound is optimal in a broader sense. Our approach allows to obtain "missing" factors in Hoeffding's inequality whose existence is motivated by the central limit theorem. The later re…
We prove deviation bounds for the random variable $\sum_{i=1}^{n} f_i(Y_i)$ in which $\{Y_i\}_{i=1}^{\infty}$ is a Markov chain with stationary distribution and state space $[N]$, and $f_i: [N] \rightarrow [-a_i, a_i]$. Our bound improves upon previously known bounds in that the dependence is on $\sqrt{a_1^2+\cdots+a_n^2}$ rather than $\max_{i}\{a_i\}\sqrt{n}.$ We also prove deviation bounds for certain types of sums of vector--valued random variables obtained from a Markov chain in a similar manner. One application includes bounding the expected value of the Schatten $\infty$-norm of a random…
We provide a systematic approach to deal with the following problem. Let $X_1,\ldots,X_n$ be, possibly dependent, $[0,1]$-valued random variables. What is a sharp upper bound on the probability that their sum is significantly larger than their mean? In the case of independent random variables, a fundamental tool for bounding such probabilities is devised by Wassily Hoeffding. In this paper we consider analogues of Hoeffding's result for sums of dependent random variables for which we have certain information on their dependency structure. We prove a result that yields concentration inequalitie…
Hoeffding's Inequality for Markov Chains under Generalized Concentrability Condition
本文提出基于积分概率度量的广义可集中性条件下马尔可夫链Hoeffding不等式框架，扩展了传统适用范围，并验证其在机器学习多类非渐近分析中的效用。
A short operator proof of Hoeffding inequalities for Markov chains
本文用两个关键观察（算子分解与秩一扰动）给出马尔可夫链加性泛函及随机矩阵Hoeffding不等式的简短算子证明，恢复并改进现有结果。
It is well known that Hoeffding's inequality has a lot of applications in the signal and information processing fields. How to improve Hoeffding's inequality and find the refinements of its applications have always attracted much attentions. An improvement of Hoeffding inequality was recently given by Hertz \cite{r1}. Eventhough such an improvement is not so big, it still can be used to update many known results with original Hoeffding's inequality, especially for Hoeffding-Azuma inequality for martingales. However, the results in original Hoeffding's inequality and its refinement one by Hertz…
A New Application of Hoeffding's Inequality Can Give Traders Early Warning of Financial Regime Change
本文提出利用 Hoeffding 不等式对交易策略表现的偏离程度进行分析，以预测金融制度转变的早期预警。
Concentration inequalities using higher moments information
针对已知随机变量前 p 阶矩的情形，本文提出更紧且具闭式形式的推广 Hoeffding 与 Bennett 不等式，并通过矩母函数新界及凸性分析加以证明。
The well-known Bennett-Hoeffding bound for sums of independent random variables is refined, by taking into account truncated third moments, and at that also improved by using, instead of the class of all increasing exponential functions, the much larger class of all generalized moment functions f such that f and f" are increasing and convex. It is shown that the resulting bounds have certain optimality properties. Comparisons with related known bounds are given. The results can be extended in a standard manner to (the maximal functions of) (super)martingales.
Hoeffding's inequality for continuous-time Markov chains
本文针对可数状态空间上不可约且正常返的连续时间马尔可夫链（CTMC），利用骨架链和增广截断逼近技术推导出专门的 Hoeffding 不等式，将其从离散时间推广至连续时间，并探讨了在一般状态空间跳跃过程上的应用。
In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) 13-30], several inequalities for tail probabilities of sums M_n=X_1+... +X_n of bounded independent random variables X_j were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. Hautes Etudes Sci. Publ. Math. 81 (1995a) 73-205] inserted certain missing factors in the bounds of two theorems. By similar factors, a third theorem was refined by Pinelis [Progress in Probability 43 (1998) 257-314] and refined (and extended) …
We obtain a Bernstein type Gaussian concentration inequality for martingales. Our inequality improves the Azuma-Hoeffding inequality for moderate deviations $x$. Following the work of McDiarmid (1989), Talagrand (1996) and Boucheron, Lugosi and Massart (2000,2003), we show that our result can be applied to the concentration of random functions, Erdös-Rényi random graph, and maxima of empirical processes. Several interesting Gaussian concentration inequalities have been obtained.
We prove a multivariate functional version of de Jong's CLT (1990) yielding that, given a sequence of vectors of Hoeffding-degenerate U-statistics, the corresponding empirical processes on $[0,1]$ weakly converge in the Skorohod space as soon as their fourth cumulants in $t=1$ vanish asymptotically and a certain strengthening of the Lindeberg-type condition is verified. As an application, we lift to the functional level the `universality of Wiener chaos' phenomenon first observed in Nourdin, Peccati and Reinert (2010).
We investigate the properties of a discrete-time martingale $\{X_m\}_{m\in \mathbb Z_{\geq 0}}$, where all differences between adjacent random variables are limited to be not more than a constant as a promise. In this situation, it is known that the Azuma-Hoeffding inequality holds, which gives an upper bound of a probability for exceptional events. The inequality gives a simple form of the upper bound, and it has been utilized for many investigations. However, the inequality is not tight. We give an explicit expression of a tight upper bound, and we show that it and the bound obtained from th…
The concentration inequality approach for normal approximation by Stein's method is generalized to the multivariate setting. We use this approach to prove a non-smooth function distance for multivariate normal approximation for standardized sums of $k$-dimensional independent random vectors $W=\sum_{i=1}^n X_i$ with an error bound of order $k^{1/2}γ$ where $γ=\sum_{i=1}^n E|X_i|^3$. For sums of locally dependent (unbounded) random vectors, we obtain a fourth moment bound which is typically of order $O_k(1/\sqrt{n})$, as well as a third moment bound which is typically of order $O_k(\log n/\sqrt…
We prove a multivariate version of Bernstein's inequality about the probability that degenerate $U$-statistics take a value larger than some number $u$. This is an improvement of former estimates for the same problem which yields an asymptotically sharp estimate for not too large numbers $u$. This paper also contains an analogous bound about the distribution of multiple Wiener-Ito integrals. Their comparison shows that our results are sharp. The proofs are based on good estimates about high moments of multiple random integrals. They are obtained by means of a diagram formula which enables us t…
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 degrees of freedom and sufficiently large non-integer "degrees of freedom" was recently proved in [10]. Here, this inequality is partly extended to smaller non-integer degrees of freedom and in particular - in a weaker form - to all infinitely divisible multivariate gamma distributions. A further m…
The pivotal set of a Boolean function
本文定义了布尔函数的 pivotal set，在输入为独立随机抛硬币时证明了关于其期望大小的基本不等式，并由此推导出 Margulis--Russo 公式及相关的偏差不等式。
We establish concentration inequalities for Lipschitz functions of dependent random variables, whose dependencies are specified by forests. We also give concentration results for decomposable functions, improving Janson's Hoeffding-type inequality for the summation of graph-dependent bounded variables. These results extend McDiarmid's bounded difference inequality to the dependent cases.
A Probability Inequality for Convolutions of MTP2-Distribution Functions
本文针对MTP2光滑分布函数的n重卷积证明了一个比高斯相关型不等式更强的概率不等式，并给出由Wishart矩阵对角线导出的多元卡方分布作为重要例子。
We use a generalization of Hoeffding's inequality to show concentration results for the free energy of disordered pinning models, assuming only that the disorder has a finite exponential moment. We also prove some concentration inequalities for directed polymers in random environment, which we use to establish a large deviations results for the end position of the polymer under the polymer measure.
We consider the stochastic integrals of multivariate point processes and study their concentration phenomena. In particular, we obtain a Bernstein type of concentration inequality through Doléans-Dade exponential formula and a uniform exponential inequality using a generic chaining argument. As applications, we obtain a upper bound for a sequence of discrete time martingales indexed by a class of functionals, and so derive the rate of convergence for nonparametric maximum likelihood estimators, which is an improvement of earlier work of van de Geer.
Let $S_N$ be the sum of vector-valued functions defined on a finite Markov chain. An analogue of the Bernstein--Hoeffding inequality is derived for the probability of large deviations of $S_N$ and relates the probability to the spectral gap of the Markov chain. Examples suggest that this inequality is better than alternative inequalities if the chain has a sufficiently large spectral gap and the function is high-dimensional.
Multivariate correlation inequalities for $P$-partitions
本文将 Fishburn、Daykin–Daykin–Paterson 及 Brightwell–Felsner–Trotter 等偏序集相关不等式推广到多元 $P$-分拆情形，证明依赖 Ahlswede–Daykin 不等式的多元推广。
We prove a new Bernstein-type inequality for the log-likelihood function of Bernoulli variables. In contrast to classical Bernstein's inequality and Hoeffding's inequality when applied to the log-likelihood, the new bound is independent of the parameters of the Bernoulli variables and therefore does not blow up as the parameters approach 0 or 1. The new inequality strengthens certain theoretical results on likelihood-based methods for community detection in networks and can be applied to other likelihood-based methods for binary data.
For a multivariate stationary process, we develop explicit representations for the finite predictor coefficient matrices, the finite prediction error covariance matrices and the partial autocorrelation function (PACF) in terms of the Fourier coefficients of its phase function in the spectral domain. The derivation is based on a novel alternating projection technique and the use of the forward and backward innovations corresponding to the predictions based on the infinite past and future, respectively. We show that such representations are ideal for studying the rates of convergence of the fini…
The Gaussian product inequality is an important conjecture concerning the moments of Gaussian random vectors. While all attempts to prove the Gaussian product inequality in full generality have been unsuccessful to date, numerous partial results have been derived in recent decades and we provide here further results on the problem. Most importantly, we establish a strong version of the Gaussian product inequality for multivariate gamma distributions in the case of nonnegative correlations, thereby extending a result recently derived by Genest and Ouimet [5]. Further, we show that the Gaussian …
Quantitative multivariate central limit theorems for general functionals of possibly non-symmetric and non-homogeneous infinite Rademacher sequences are proved by combining discrete Malliavin calculus with the smart path method for normal approximation. In particular, a discrete multivariate second-order Poincaré inequality is developed. As a first application, the normal approximation of vectors of subgraph counting statistics in the Erdős-Rényi random graph is considered. In this context, we further specialize to the normal approximation of vectors of vertex degrees. In a second application …
The stability of the multivariate geometric Brownian motion as a bilinear matrix inequality problem
本文研究多元几何布朗运动原点的全局渐近稳定性，通过构造Lyapunov函数并将稳定性条件转化为BMI可行性问题，适用于多种具体模型。
Let $μ$ be a Gaussian measure (say, on ${\bf R}^n$) and let $K, L \subset {\bf R}^n$ be such that K is convex, $L$ is a "layer" (i.e. $L = \{x : a \leq < x,u > \leq b \}$ for some $a$, $b \in {\bf R}$ and $u \in {\bf R}^n$) and the centers of mass (with respect to $μ$) of $K$ and $L$ coincide. Then $μ(K \cap L) \geq μ(K) \cdot μ(L)$. This is motivated by the well-known "positive correlation conjecture" for symmetric sets and a related inequality of Sidak concerning confidence regions for means of multivariate normal distributions. The proof uses an apparently hitherto unknown estimate for the …
This is a substantially generalized version of the preprint arXiv:1105.4214 by Lifshits and Tyurin. We prove that for any pair of i.i.d. random vectors $X, Y$ in $R^n$ and any real-valued continuous negative definite function $g: R^n\to R$ the inequality $$ E g(X-Y) \le E g(X+Y)$$ holds. In particular, for $a \in (0,2]$ and the Euclidean norm $|.|$ one has $$ E |X-Y|^a \le E |X+Y|^a. $$ The latter inequality is due to A. Buja et al. (Ann. Statist., 1994} where it is used for some applications in multivariate statistics. We show a surprising connection with bifractional Brownian motion and prov…
For a real-valued measurable function $f$ and a nonnegative, nondecreasing function $ϕ$, we first obtain a Chebyshev type inequality which provides an upper bound for $\displaystyle ϕ(λ_{1}) μ(\{x \in Ω: f(x) \geq λ_{1} \}) + \sum_{k=2}^{n}\left(ϕ(λ_{k})- ϕ(λ_{k-1})\right) μ(\{x \in Ω: f(x) \geq λ_{k}\}) ,$ where $0 < λ_1 < λ_2 \cdots λ_n < \infty$. Using this, generalizations of a few concentration inequalities such as Markov, reverse Markov, Bienaymé-Chebyshev, Cantelli and Hoeffding inequalities are obtained.
In this paper, we provide a new framework to obtain the generalization bounds of the learning process for domain adaptation, and then apply the derived bounds to analyze the asymptotical convergence of the learning process. Without loss of generality, we consider two kinds of representative domain adaptation: one is with multiple sources and the other is combining source and target data. In particular, we use the integral probability metric to measure the difference between two domains. For either kind of domain adaptation, we develop a related Hoeffding-type deviation inequality and a symmetr…
The software Randentropy is designed to estimate inequality in a random system where several individuals interact moving among many communities and producing dependent random quantities of an attribute. The overall inequality is assessed by computing the Random Theil's Entropy. Firstly, the software estimates a piecewise homogeneous Markov chain by identifying the changing-points and the relative transition probability matrices. Secondly, it estimates the multivariate distribution function of the attribute using a copula function approach and finally, through a Monte Carlo algorithm, evaluates…
Time-Uniform Self-Normalized Concentration for Vector-Valued Processes
本文为满足 sub-$ψ$ 尾部条件的多变量过程构造了通用的自归一化集中不等式，推导出紧致的迭代对数上定律，并应用于在线线性回归和自回归建模等统计任务。
We present arguments in favour of the inequalities $var(X_n^2|X \in B_v(ρ)) \le 2λ_n E[X_n^2|X \in B_v(ρ)]$, where $X \sim N_v(0,Λ)$ is a normal vector in $v\ge 1$ dimensions, with zero mean and covariance matrix $Λ= \diag(λ)$, and $B_v(ρ)$ is a centered $v$-dimensional Euclidean ball of square radius $ρ$. Such relations lie at the heart of an iterative algorithm, proposed in ref. [1] to perform a reconstruction of $Λ$ from the covariance matrix of $X$ conditioned to $B_v(ρ)$. In the regime of strong truncation, i.e. for $ρ\lesssim λ_n$, the above inequality is easily proved, whereas it become…
Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that random functions are near their means. Of particular importance is the case where f(X) is a function of independent random variables X=(X_1, ..., X_n). Here the well known bounded differences inequality (also called McDiarmid's or Hoeffding-Azuma inequality) establishes sharp concentration if the function f does not depend too much on any of the variables. One attractive feature is that it relies on a very simple Lipschitz condition (L): it suffices to show that |f(…
Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality
本文通过 Hardy 和 Krause 意义下的多变量变分函数与有限总变分有号测度之间的对应关系，推导出非均匀测度下的广义 Koksma-Hlawka 不等式，并探讨了将其应用于重要采样和可处理性理论的应用，以及将均匀测度下的低偏差序列转换为一般测度下低偏差序列的限制。
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