Xiequan Fan, Ion Grama, Quansheng Liu
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.
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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 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.
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 explicitly construct the supermartingale version of the Fr{é}chet-Hoeffding coupling in the setting with infinitely many marginal constraints. This extends the results of Henry-Labordere et al. obtained in the martingale setting. Our construction is based on the Markovian iteration of one-period optimal supermartingale couplings. In the limit, as the number of iterations goes to infinity, we obtain a pure jump process that belongs to a family of local L{é}vy models introduced by Carr et al. We show that the constructed processes solve the continuous-time supermartingale optimal transport pr…
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…
A tight upper bound is given on the distribution of the maximum of a supermartingale. Specifically, it is shown that if $Y$ is a semimartingale with initial value zero and quadratic variation process $[Y,Y]$ such that $Y + [Y,Y]$ is a supermartingale, then the probability the maximum of $Y$ is greater than or equal to a positive constant $a$ is less than or equal to $1/(1+a).$ The proof makes use of the semimartingale calculus and is inspired by dynamic programming.
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…
The extended Ville's inequality for nonintegrable nonnegative supermartingales
该研究继Robbins的工作后，提出了无需可积性与有限性的非负上鞅扩展理论，推导了扩展Ville不等式与适用于σ-有限混合的扩展混合方法，并揭示了其在序贯统计中的应用价值。
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。
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。
Freedman's inequality is a supermartingale counterpart to Bennett's inequality. This result shows that the tail probabilities of a supermartingale is controlled by the quadratic characteristic and a uniform upper bound for the supermartingale difference sequence. Replacing the quadratic characteristic by $\textrm{H}_k^y:= \sum_{i=1}^k\left(\mathbf{E}(ξ_i^2 |\mathcal{F}_{i-1}) +ξ_i^2\textbf{1}_{\{|ξ_i|> y\}}\right),$ Dzhaparidze and van Zanten (\emph{Stochastic Process. Appl.}, 2001) have extended Freedman's inequality to martingales with unbounded differences. In this paper, we prove that $\te…
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.
We give a simple inequality for the sum of independent bounded random variables. This inequality improves on the celebrated result of Hoeffding in a special case. It is optimal in the limit where the sum tends to a Poisson random variable.
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) …
A Generalisation of Ville's Inequality to Monotonic Lower Bounds and Thresholds
该研究将经典Ville不等式的常数下界与阈值推广为单调曲线，给出保持在递减下界曲线上方的上鞅超过递增阈值曲线的概率紧界，并据此推导出有限时间版本重对数律。
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 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.
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.
The Exact Ville Identity: From the Absorbing Case to the General Law with an Application to E-Values
研究非负上鞅首次达到大于1的水平的概率，通过证明精确恒等式，将维勒不等式的松弛部分完全分解为过冲、损失和生存部分，给出证明并应用于多种示例及序贯e值检验。
We present novel martingale concentration inequalities for martingale differences with finite Orlicz-$ψ_α$ norms. Such martingale differences with weak exponential-type tails scatters in many statistical applications and can be heavier than sub-exponential distributions. In the case of one dimension, we prove in general that for a sequence of scalar-valued supermartingale difference, the tail bound depends solely on the sum of squared Orlicz-$ψ_α$ norms instead of the maximal Orlicz-$ψ_α$ norm, generalizing the results of Lesigne & Volný (2001) and Fan et al. (2012). In the multidimensional ca…
Let $(S_0,S_1,...)$ be a supermartingale relative to a nondecreasing sequence of $σ$-algebras $H_{\le0},H_{\le1},...$, with $S_0\le0$ almost surely (a.s.) and differences $X_i:=S_i-S_{i-1}$. Suppose that $X_i\le d$ and $\mathsf {Var}(X_i|H_{\le i-1})\le σ_i^2$ a.s. for every $i=1,2,...$, where $d>0$ and $σ_i>0$ are non-random constants. Let $T_n:=Z_1+...+Z_n$, where $Z_1,...,Z_n$ are i.i.d. r.v.'s each taking on only two values, one of which is $d$, and satisfying the conditions $\mathsf {E}Z_i=0$ and $\mathsf {Var}Z_i=σ^2:=\frac{1}{n}(σ_1^2+...+σ_n^2)$. Then, based on a comparison inequality …
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(…
Bounded Difference Concentration for Infinitely Exchangeable Sequences with Applications to AI Benchmark Uncertainty
通过de Finetti测度分解有界差分函数的偏差，提出有效方差代理的集中不等式，并证明零和线性对比中潜在混合项完全抵消，应用于AI基准如MMLU的不确定性量化。
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