Deviation Inequalities Research Articles

Optimal asymptotic quadratic error of nonparametric regression function estimates for a continuous-time process from sampled-data

For different classes of deterministic and random sampling (t k ), we establish the asymptotic expressions for the bias and the variance of the estimate r n(x) based on sampled data for the regression function r(x) = E(Y t X t = x) of unbounded continuous-time processes (not necessarily stationary). Under mild mixing conditions, we show that r n (x) has exactly the same asymptotic quadratic error as in the i.i.d. case. In order to prove this result, we use some large deviations inequalities for mixing processes.

Near-optimal, distributed edge colouring via the nibble method

We give a distributed randomized algorithm for graph edge colouring. Let G be a Δ-regular graph with n nodes. Here we prove: • • If ε>0 is fixed and Δ⪢ log n, the algorithm almost always colours G with (1 + ε) Δ colours in time O( log n). • • If s>0 is fixed, there exists a positive constant k such that if Δ⪢ log k n, the algorithm almost always colours G with Δ + Δ/ log s n colours in time O( log n + log s n log log n). By “almost always” we mean that the algorithm may either use more than the claimed number of colours or run longer than the claimed time, but that the probability that either of these sorts of failure occurs can be made arbitrarily close to 0. The algorithm is based on the nibble method, a probabilistic strategy introduced by Vojtěch Rödl. The analysis makes use of a powerful large deviation inequality for functions of independent random variables.

Martingales, Poincaré Type Inequalities, and Deviation Inequalities

Using martingale techniques we will prove several deviation inequalities for diffusion processes in a compact Riemannian manifold and Lévy processes in euclidean space. We also deduce deviation inequalities from Poincaré type inequalities in the abstract setting of Dirichlet forms. We thus obtain, in a unified way, the theorems of Alon–Milman, Gromov–Milman, and a deviation inequality for the Laplace disribution.

A Large Deviation Inequality for Functions of Independent, Multi-Way Choices

Often when analysing randomized algorithms, especially parallel or distributed algorithms, one is called upon to show that some function of many independent choices is tightly concentrated about its expected value. For example, the algorithm might colour the vertices of a given graph with two colours and one would wish to show that, with high probability, very nearly half of all edges are monochromatic.The classic result of Chernoff [3] gives such a large deviation result when the function is a sum of independent indicator random variables. The results of Hoeffding [5] and Azuma [2] give similar results for functions which can be expressed as martingales with a bounded difference property. Roughly speaking, this means that each individual choice has a bounded effect on the value of the function. McDiarmid [9] nicely summarized these results and gave a host of applications. Expressed a little differently, his main result is as follows.

Fast Distributed Algorithms for Brooks-Vizing Colourings (Extended Abstract)

Fast Distributed Algorithms for Brooks-Vizing Colourings (Extended Abstract)

Parametric rates of nonparametric estimators and predictors for continuous time processes

We show that local irregularity of observed sample paths provides additional information which allows nonparametric estimators and predictors for continuous time processes to reach parametric rates in mean square as well as in a.s. uniform convergence. For example, we prove that under suitable conditions the kernel density estimator $f_T$ associated with the observed sample path $(X_t, 0 \leq t \leq T)$ satisfies $$\sup_{x \epsilon \mathbb{R}}|f_T(x) - f(x)| = o(\ln_k T(\frac{\ln T}{T})^{1/2}) \quad \text{a.s.}, k \geq 1$$ where f denotes the unknown marginal density of the stationary process and where $\ln_k$ denotes the kth iterated logarithm. The proof uses a special Borel-Cantelli lemma for continuous time processes together with a sharp large deviation inequality. Furthermore the parametric rate obtained in (1) is preserved by using a suitable sampling scheme.

On Talagrand's deviation inequalities for product measures

We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. Talagrand in the recent years. Our method is based on functional inequalities of Poincare and logarithmic Sobolev type and iteration of these inequalities. In particular, we establish with theses tools sharp deviation inequalities from the mean on norms of sums of independent random vectors and empirical processes. Concentration for the Hamming distance may also be deduced from this approach.

Deviation inequalities for continuous martingales

We consider a broad class of continuous martingales whose local modulus of continuity is in some sense deterministic. We show that such martingales have Gaussian probability tails, provided we appropriately normalize them by their quadratic variation. As other applications of our methods, we provide energy inequalities and prove a new sufficient condition for the joint continuity of continuous additive functionals of Brownian motion indexed by their Revuz measures.

The Wills functional and Gaussian processes

The Wills functional from the theory of lattice point enumeration can be adapted to produce the following exponential inequality for zero-mean Gaussian processes: $$E \exp [\sup_t (X_t - (1/2) \sigma_t^2)] \leq \exp (E \sup_t X_t).$$ An application is a new proof of the deviation inequality for the supremum of a Gaussian process above its mean: $$P(\sup_t X_t - E \sup_t X_t \geq a) \leq \exp (-\frac{(1/2) \alpha^2}{\sigma^2}),$$ where $a > 0$ and $\sigma^2 = \sup_t \sigma_t^2$.

Near-Optimal, Distributed Edge Colouring via the Nibble Method

We give a distributed randomized algorithm to edge colour a network. Let G be a graph<br />with n nodes and maximum degree Delta. Here we prove:<br /> If Delta = Omega(log^(1+delta) n) for some delta > 0 and lambda > 0 is fixed, the algorithm almost always<br />colours G with (1 + lambda)Delta colours in time O(log n).<br /> If s > 0 is fixed, there exists a positive constant k such that if Delta = omega(log^k n), the algorithm almost always colours G with Delta + Delta / log^s n = (1+o(1))Delta colours in time<br />O(logn + log^s n log log n).<br />By "almost always" we mean that the algorithm may fail, but the failure probability can be<br />made arbitrarily close to 0.<br />The algorithm is based on the nibble method, a probabilistic strategy introduced by<br />Vojtech R¨odl. The analysis makes use of a powerful large deviation inequality for functions<br />of independent random variables.

Bounding Ramsey numbers through large deviation inequalities

AbstractWe develop a new approach for proving lower bounds for various Ramsey numbers, based on using large deviation inequalities. This approach enables us to obtain the bounds for the off‐diagonal Ramsey numbers R(Kr, Kk), r ≤ k, that match the best known bounds, obtained through the local lemma. We discuss also the bounds for a related Ramsey‐type problem and show, for example, that there exists a K4‐free graph G on n vertices in which every cn3/5 log1/2 n vertices span a copy of K3. © 1995 John Wiley & Sons, Inc.

Bernstein inequalities for a class of random variables

We prove a deviation inequality for sums of i.i.d. random variables.

Bernstein-type large deviations inequalities for partial sums of strong mixing processes

By using an approximation of partial sums of aprocess by independent r.v.'s we obtain large deviation inequalities. In particular we show that if is a centered real valued process, if there exist constants m and M such that and if 1≥p n ≥n/2 then, for every ϵ>0, we have where (α(p), p ≥ 1) denote ihe strong mixing coefficients of (Xt). This inequality is simpler and more explicit than those which are actually available in the literature. Similarly we obtain an inequality for processes satisfying Cramer's conditions. Applications to the rate of convergence in the law of large numbers and to nonparametric regression with dependent errors are given.

Parameter estimation in linear filtering

Suppose on a probability space (Ω, F, P), a partially observable random process ( x t , y t ), t ≥ 0; is given where only the second component ( y t ) is observed. Furthermore assume that ( x t , y t ) satisfy the following system of stochastic differential equations driven by independent Wiener processes ( W 1( t)) and ( W 2( t)): dx t =− βx t dt+ dW 1( t), x 0=0, dy t = αx t dt+ dW 2( t), y 0=0; α, β∞( a, b), a>0. We prove the local asymptotic normality of the model and obtain a large deviation inequality for the maximum likelihood estimator (m.l.e.) of the parameter θ = (α, β). This also implies the strong consistency, efficiency, asymptotic normality and the convergence of moments for the m.l.e. The method of proof can be easily extended to obtain similar results when vector valued instead of one-dimensional processes are considered and θ is a k-dimensional vector.

Some deviation inequalities

We introduce a concentration property for probability measures onR n , which we call Property (τ); we show that this property has an interesting stability under products and contractions (Lemmas 1, 2, 3). Using property (τ), we give a short proof for a recent deviation inequality due to Talagrand. In a third section, we also recover known concentration results for Gaussian measures using our approach.

Applications of Deviation Inequalities on Finite Metric Sets

Mathematische NachrichtenVolume 153, Issue 1 p. 33-41 Article Applications of Deviation Inequalities on Finite Metric Sets Jesús Bastero, Jesús Bastero Departamento of Matematicas Facultad de Ciencias Universidad de Zaragoza 50009 Zaragoza SpainSearch for more papers by this authorJulio Bernués, Julio Bernués Departamento of Matematicas Facultad de Ciencias Universidad de Zaragoza 50009 Zaragoza Spain The two named authors are partially supported by Grant DGICYT PS 87-0059. The second named author is also supported by D.G.A. (Spain)Search for more papers by this author Jesús Bastero, Jesús Bastero Departamento of Matematicas Facultad de Ciencias Universidad de Zaragoza 50009 Zaragoza SpainSearch for more papers by this authorJulio Bernués, Julio Bernués Departamento of Matematicas Facultad de Ciencias Universidad de Zaragoza 50009 Zaragoza Spain The two named authors are partially supported by Grant DGICYT PS 87-0059. The second named author is also supported by D.G.A. (Spain)Search for more papers by this author First published: 1991 https://doi.org/10.1002/mana.19911530104Citations: 2 AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume153, Issue11991Pages 33-41 RelatedInformation

Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures

Let (X i ,Y i ) ∈ℤ d , be independent identically distributed random variables with arbitrary distribution. We show that, for almost every(Y i ) i , the conditional law of the empirical field given(Y i ) i satisfies to large deviation inequalities. This applies to the study of Gibbs measures with random interaction, in the case of some mean-field models as well as of short range summable interaction. We show that the pressure is nonrandom, and is given by a variational formula. These random Gibbs measures have the same large deviation rate, which does not depend on the particular realization of the interaction; their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula.

A Sharp Deviation Inequality for the Stochastic Traveling Salesman Problem

Let $T_n$ denote the length of the shortest closed path connecting $n$ random points uniformly distributed over the unit square. We prove that for some number $K$, we have, for all $t \geq 0$, $P(|T_n - E(T_n)| \geq t) \leq K \exp(-t^2/K).$

Probability Inequalities for Sums of Independent Random Variables

Probability Inequalities for Sums of Independent Random Variables

Prescribing a System of Random Variables by Conditional Distributions

Prescribing a System of Random Variables by Conditional Distributions