Let $(X_{i}, \mathcal{F}_{i})_{i\geq 1}$ be a sequence of supermartingale differences and let $S_k=\sum_{i=1}^k X_i$. We give an exponential moment condition under which $P(\max_{1\leq k \leq n} S_k \geq n)=O(\exp\{-C_1 n^α\}),$ $n\rightarrow \infty,$ where $α\in (0, 1)$ is given and $C_{1}>0$ is a constant. We also show that the power $α$ is optimal under the given condition. In particular, when $α=1/3$, we recover an inequality of Lesigne and Volný.
In this work, Bernstein's concentration inequalities for squared integrable matrix-valued discrete-time martingales are obtained. Based on Lieb's theory and Bernstein's condition, a suitable supermartingale can be constructed. Our proof is largely based on this new exponential supermartingale, Freedman's method, and Doob's stopping theorem. Our result can be regarded as an extension of Tropp's work (ECP, 2012).
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…
The paper is devoted to establishing some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett, Freedman, de la Peña, Pinelis and van de Geer. Moreover, our concentration inequalities also improve some known inequalities for sums of independent random variables. Applications associated with linear regressions, autoregressive processes and branching processes are provided. In particular, an interesting application of {de la Peña's} inequality to self-normalized deviations is also provided.
We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of a strictly stationary orthomartingale random field. These inequalities can be used to establish complete convergence of normalized maxima of partial sums.
Random matrices have played an important role in many fields including machine learning, quantum information theory and optimization. One of the main research focuses is on the deviation inequalities for eigenvalues of random matrices. Although there are intensive studies on the large-deviation inequalities for random matrices, only a few of works discuss the small-deviation behavior of random matrices. In this paper, we present the small-deviation inequalities for the largest eigenvalues of sums of random matrices. Since the resulting inequalities are independent of the matrix dimension, they…
This note is concerned with lower tail estimates for product measures. Some improved deviation inequalities are obtained for functions satisfying some regularity and monotonicity assumptions. The arguments are based on semigroup interpolation together with Harris's negative association inequality and hypercontractive estimates.
Let $\left \{ Z_{n}, n\ge 0 \right \}$ be a supercritical branching process in an independent and identically distributed random environment $ξ=\left ( ξ_{n} \right )_{n\geq 0} $. In this paper, we get some deviation inequalities for $\ln \left (Z_{n+n_{0} } / Z_{n_{0} } \right ).$ And some applications are given for constructing confidence intervals.
Using changes of probability measure developed by \mbox{Grama} and Haeusler (Stochastic Process.\ Appl., 2000), we obtain two generalizations of the deviation inequalities of Lanzinger and Stadtmüller (Stochastic Process.\ Appl., 2000) and Fuk and Nagaev (Theory Probab. Appl., 1971) to the case of martingales. Our inequalities recover the best possible decaying rate of independent case. Applications to linear regressions and weak invariance principles for martingales are provided.
We obtain moderate deviations theorems and exponential (Bernstein type) concentration inequalities for "nonconventional" sums of the form $S_N=\sum_{n=1}^N (F(ξ_{q_1(n)},ξ_{q_2(n)},...,ξ_{q_\ell(n)})-\bar F)$.
The purpose of this note is to show how simple Optimal Transport arguments, on the real line, can be used in Superconcentration theory. This methodology is efficient to produce sharp non-asymptotic variance bounds for various functionals (maximum, median, $l^p$ norms) of standard Gaussian random vectors in $\R^n$. The flexibility of this approach can also provide exponential deviation inequalities reflecting preceding variance bounds. As a further illustration, usual laws from Extreme theory and Coulomb gases are studied.
Poisson-type deviation inequalities for curved continuous-time Markov chains
本文对Wasserstein曲率或$Γ$ -曲率有下界的连续时间马尔可夫链，给出了新的Poisson型偏差不等式.虽然这两个曲率对于黎曼流形上的布朗运动是等价的，但它们在离散情况下是不可比较的，并且产生不同的偏差界。在生灭过程的情形下，我们给出了相应的生成元的转移率使这类曲率有界的条件，并推广了[Ané，C.和Ledoux，M.关于图上连续时间随机游动的对数Sobolev不等式。可能吧Theory Related Fields 116（2000）573--602]，用于连续时间随机游动，被看作零曲率中的模型。这些尾部估计的一些应用给出了布朗驱动的Ornstein-Uhlenbeck过程和$M/M/1$队列。
A survey is given of some Chernoff type bounds for the tail probabilities P(X-EX > a) and P(X-EX < a) when X is a random variable that can be written as a sum of indicator variables that are either independent or negatively related. Most bounds are previously known and some comparisons are made. This paper was written in 1994, but was never published because I had overlooked some existing papers containing some of the inequalities. Because of some recent interest in one of the inequalities, which does not seem to be published anywhere else, it has now been lightly edited and made available her…
Let $\{X_i,i\geq1\}$ be a sequence of negatively associated random variables, and let $\{X_i^\ast,i\geq 1\}$ be a sequence of independent random variables such that $X_i^\ast$ and $X_i$ have the same distribution for each $i$. Denote by $S_k=\sum_{i=1}^{k}X_i$ and $S_k^\ast=\sum_{i=1}^{k}X_i^\ast$ for $k\geq 1$. The well-known results of Shao \cite{Shao2000} sates that $\mathbb{E}f(S_n)\leq \mathbb{E}f(S_n^\ast)$ for any nondecreasing convex function. Using this very strong property, we obtain a large variety of deviation inequalities for $S_n$
Let $(ξ_i)_{i=1,...,n}$ be a sequence of independent and symmetric random variables. We consider the upper bounds on tail probabilities of self-normalized deviations $$ \mathbf{P} \Big( \max_{1\leq k \leq n} \sum_{i=1}^{k} |ξ_i|\big/ \big(\sum_{i=1}^{n} |ξ_i|^β\big)^{1/β} \geq x \Big) $$ for $x>0$ and $β>1.$ Our bound is the best that can be obtained from the Bernstein inequality under the present assumption. An application to Student's $t$-statistics is also given.
We consider Markov chains which are polynomially mixing, in a weak sense expressed in terms of the space of functions on which the mixing speed is controlled. In this context, we prove polynomial large and moderate deviations inequalities. These inequalities can be applied in various natural situations coming from probability theory or dynamical systems. Finally, we discuss examples from these various settings showing that our inequalities are sharp.
Let $(X_i, \mathcal{F}_i)_{i\geq1}$ be a martingale difference sequence in a smooth Banach space. Let $S_n=\sum_{i=1}^nX_i, n\geq 1,$ be the partial sums of $(X_i, \mathcal{F}_i)_{i\geq 1}$. We give upper bounds on the quantity $\mathbb{P}\left(\max_{1\leq k\leq n}\lVert S_k\rVert>nx\right)$ in terms of $ n\geq 1$ and $x>0$ in two different situations: when the martingale differences have uniformly bounded exponential moments and when the decay of the tail of the increments is polynomial.
In this paper, we establish a large deviation principle for the solutions to the stochastic heat equations with logarithmic nonlinearity driven by Brownian motion, which is neither locally Lipschitz nor locally monotone. Nonlinear versions of Gronwall's inequalities and Log-Sobolev inequalities play an important role.
We obtain large and moderate deviation estimates, as well as concentration inequalities, for a class of nonuniformly expanding maps with stretched exponential decay of correlations. In the large deviation regime, we also exhibit examples showing that the obtained upper bounds are essentially optimal.
Suppose we observe an infinite series of coin flips $X_1,X_2,\ldots$, and wish to sequentially test the null that these binary random variables are exchangeable. Nonnegative supermartingales (NSMs) are a workhorse of sequential inference, but we prove that they are powerless for this problem. First, utilizing a geometric concept called fork-convexity (a sequential analog of convexity), we show that any process that is an NSM under a set of distributions, is also necessarily an NSM under their "fork-convex hull". Second, we demonstrate that the fork-convex hull of the exchangeable null consists…
Motivated by Talagrand's conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and in particular its continuous analogue involving regularization properties of the Ornstein-Uhlenbeck semigroup acting on in-tegrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure.
We consider a class of non-homogeneous Markov chains, that contains many natural examples. Next, using martingale methods, we establish some deviation and moment inequalities for separately Lipschitz functions of such a chain, under moment conditions on some dominating random variables.
We introduce a class of Markov chains, that contains the model of stochastic approximation by averaging and non-averaging. Using martingale approximation method, we establish various deviation inequalities for separately Lipschitz functions of such a chain, with different moment conditions on some dominating random variables of martingale differences.Finally, we apply these inequalities to the stochastic approximation by averaging and empirical risk minimisation.
Our goal is to establish large deviations and concentration inequalities for the maximum likelihood estimator of the drift parameter of the Ornstein-Uhlenbeck process without tears. We propose a new strategy to establish large deviation results which allows us, via a suitable transformation, to circumvent the classical difficulty of non-steepness. Our approach holds in the stable case where the process is positive recurrent as well as in the unstable and explosive cases where the process is respectively null recurrent and transient. Notwithstanding of this trichotomy, we also provide new conce…
We derive two upper bounds for the probability of deviation of a vector-valued Lipschitz function of a collection of random variables from its expected value. The resulting upper bounds can be tighter than bounds obtained by a direct application of a classical theorem due to Bobkov and Götze.
We provide a stochastic interpretation of non-commutative Dirichlet forms in the context of quantum filtering. For stochastic processes motivated by quantum optics experiments, we derive an optimal finite time deviation bound expressed in terms of the non-commutative Dirichlet form. Introducing and developing new non-commutative functional inequalities, we deduce concentration inequalities for these processes. Examples satisfying our bounds include tensor products of quantum Markov semigroups as well as Gibbs samplers above a threshold temperature.
We present order of magnitude estimates for the quantiles of non-negative linear combinations of non-negative random variables, as well as deviation inequalities for general linear combinations of independent random variables, under the assumption that all random variables satisfy the same power-type tail bound on $\mathbb{P}\{\left\vert X_i\right\vert>t\}$ of the form $t^{-q}$, $t^{-q/2}$ or $t^{-q/2}(\ln t)^{q/2}$, for $q>2$. The third type is applicable in the nonlinear setting. In the situations we consider, these results improve on classical estimates of Nagaev.
