Large Deviation Inequality Research Articles

Large Deviations for the Graph Distance in Supercritical Continuum Percolation

Denote the Palm measure of a homogeneous Poisson processHλwith two points 0 andxby P0,x. We prove that there exists a constant μ ≥ 1 such that P0,x(D(0,x) / μ||x||2∉ (1 − ε, 1 + ε) | 0,x∈C∞) exponentially decreases when ||x||2tends to ∞, whereD(0,x) is the graph distance between 0 andxin the infinite componentC∞of the random geometric graphG(Hλ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.

Large Deviations for the Graph Distance in Supercritical Continuum Percolation

Denote the Palm measure of a homogeneous Poisson process Hλ with two points 0 and x by P0,x. We prove that there exists a constant μ ≥ 1 such that P0,x(D(0, x) / μ||x||2 ∉ (1 − ε, 1 + ε) | 0, x ∈ C∞) exponentially decreases when ||x||2 tends to ∞, where D(0, x) is the graph distance between 0 and x in the infinite component C∞ of the random geometric graph G(Hλ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.

Marcinkiewicz–Zygmund inequality for nonnegative [formula omitted]-demimartingales and related results

In this paper the Marcinkiewicz–Zygmund inequality for mean zero independent random variables is extended to the case of nonnegative N -demimartingales. This result can be used for obtaining large deviation inequalities for nonnegative N -demimartingales as well as related complete convergence results.

Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations

The estimate of the probability of the large deviation or the statistical randomfieldisthekeytoensuretheconvergenceofmomentsoftheassociatedestima- tor, and it also plays an essential role to prove mathematical validity of the asymptotic expansion of the estimator. For non-linear stochastic processes, it involves technical difficultiestoshowastandardexponentialtypeestimate;besides,itisnotnecessaryfor these purposes. In this paper, we propose a polynomial-type large deviation inequality which is easily verified by the L p -boundedness of certain functionals; usually they are simple additive functionals. We treat a statistical random field with multi-grades and discuss M and Bayesian type estimators. As an application, we show the behavior of those estimators, including convergence of moments, for the statistical random field in the quasi-likelihood analysis of the stochastic differential equation that is possibly multi-dimensional and non-linear. The results are new even for stochastic differential equations, while they obviously apply to other various statistical models.

Approximate self-weighted LAD estimation of discretely observed ergodic Ornstein-Uhlenbeck processes

We consider drift estimation of a discretely observed Ornstein-Uhlenbeck process driven by a possibly heavy-tailed symmetric Lévy process with positive activity index β. Under an infill and large-time sampling design, we first establish an asymptotic normality of a self-weighted least absolute deviation estimator with the rate of convergence being $\sqrt{n}h_{n}^{1-1/\beta}$, where n denotes sample size and hn>0 the sampling mesh satisfying that hn→0 and nhn→∞. This implies that the rate of convergence is determined by the most active part of the driving Lévy process; the presence of a driving Wiener part leads to $\sqrt{nh_{n}}$, which is familiar in the context of asymptotically efficient estimation of diffusions with compound Poisson jumps, while a pure-jump driving Lévy process leads to a faster one. Also discussed is how to construct corresponding asymptotic confidence regions without full specification of the driving Lévy process. Second, by means of a polynomial type large deviation inequality we derive convergence of moments of our estimator under additional conditions.

Large deviation inequalities for supermartingales and applications to directed polymers in a random environment

We first show exponential large deviation inequalities for supermartingales, which are optimal in the independent and identically distributed (iid) case. We then apply them to establish new exponential concentration inequalities for the free energy of directed polymers in random environment; as consequences we obtain upper bounds of its rate of convergences (in probability, almost surely, and in L p ), and give an expression for the free energy in terms of free energies of some multiplicative cascades. To cite this article: Q. Liu, F. Watbled, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

A large deviation inequality for vector functions on finite reversible Markov Chains

Let SN 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 SN 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.

Large deviations for the chemical distance in supercritical Bernoulli percolation

The chemical distance D(x,y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behaviour of this random metric, and we prove that, for an appropriate norm $\mu$ depending on the dimension and the percolation parameter, the probability of the event \[\biggl\{0\leftrightarrow x,\frac{D(0,x)}{\mu(x)}\notin (1-\epsilon, 1+\epsilon) \biggr\}\] exponentially decreases when $\|x\|_1$ tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.

Ideal denoising for signals in sub-Gaussian noise

Donoho and Johnstone introduced an algorithm and supporting inequality that allows the selection of an orthonormal basis for optimal denoising. The present paper concentrates in extending and improving this result, the main contribution is to incorporate a wider class of noise vectors. The class of strict sub-Gaussian random vectors allow us to obtain large deviation inequalities in a uniform way over all basis in a given library. The results are obtained maintaining the algorithmic properties of the original results.

A noncommutative version of the John-Nirenberg theorem

We prove a noncommutative version of the John-Nirenberg theorem for nontracial filtrations of von Neumann algebras. As an application, we obtain an analogue of the classical large deviation inequality for elements of the associated BMO space.

On the Asymptotic Redundancy of Lossless Block Coding With Two Codeword Lengths

With the additional constraint of requiring only two codeword lengths, lossless codes of blocks of size n generated by stationary memoryless binary sources are studied. For arbitrary /spl delta/>0, classical large-deviation inequalities imply the existence of codes attaining an expected redundancy of the order O(n/sup -1/2+/spl delta//). It is shown that it is not possible to construct lossless codes with two codeword lengths having rate of order better or equal to O(n/sup -1/2/).

Approximating the independence number and the chromatic number in expected polynomial time

The independence number of a graph and its chromatic number are known to be hard to approximate. Due to recent complexity results, unless coRP = NP, there is no polynomial time algorithm which approximates any of these quantities within a factor of n 1−∈ for graphs on n vertices. We show that the situation is significantly better for the average case. For every edge probability p = p(n) in the range n −1/2+∈ ≤ p ≤ 3/4, we present an approximation algorithm for the independence number of graphs on n vertices, whose approximation ratio is O((np)1/2/log n) and whose expected running time over the probability space G(n, p) is polynomial. An algorithm with similar features is described also for the chromatic number. A key ingredient in the analysis of both algorithms is a new large deviation inequality for eigenvalues of random matrices, obtained through an application of Talagrand's inequality.

Nonnegative piecewise linear histograms*

We introduce a modified version ƒnof the piecewiss linear hisiugrimi uf Beirlant et al. (1998) which is a true probability density, i.e., ƒn[d] 0 and [d]ƒn=1. We prove that ƒnestimates the underlying densitv ƒ strongly consistently in the L1mmn, derive large deviation inequalities for the t\\ error \\ƒn- f\\ and prove that £||/"-/|| tends to zero with the rate n -1\\3, We also show that the derivative lf'n estimates consistently in ine expected Lx error the derivative/ of sufficiently smooth density and evaluate the rate of convergence n-i/5 for Epf'n -f'% The estimator/" thus enables to approximate/in the Besov space with a guaranteed rate of convergence. Optimization of the smoothing parameter is also studied. The theoretical or experimentally approximated values of the expected errors E\\\\ƒn- f\\\\ and E||2ƒ'n-ƒ' are compared with tiie errors aCiiieveu u-y t"e histogram of Beirlant et ah, and other nonparametric methods.

Large deviations inequalities for partial sums of random measures

We give large deviation inequalities for partial sums of random measures and for the Prokhorov metric, with applications to Poisson, Cox and compound point processes.

On piecewise linear density estimators

We study piecewise linear density estimators from the L1 point of view: the frequency polygons investigated by Scott (1985) and Jones et al. (1997), and a new piecewise linear histogram. In contrast to the earlier proposals, a unique multivariate generalization of the new piecewise linear histogram is available. All these estimators are shown to be universally L1 strongly consistent. We derive large deviation inequalities. For twice differentiable densities with compact support their expected L1 error is shown to have the same rate of convergence as have kernel density estimators. Some simulated examples are presented.

Large deviations inequalities for the maximum likelihood estimator and the Bayes estimators in nonlinear stochastic differential equations

Exponential bounds on the large deviation probability of the maximum likelihood estimator and the Bayes estimators of the parameter appearing nonlinearly in the drift coefficient of homogeneous Itô's stochastic differential equations are obtained under some regularity conditions.

A new large deviation inequality for U-statistics of order 2

We prove a new large deviation inequality with applications when projecting a density on a wavelet basis.

Exponential-type inequalities for martingale difference sequences. Application to nonparametric regression estimation

We state large deviation inequalities for the maxima of partial sums of mar¬tingale difference sequences. Some implications of our inequalities in stating the rate of convergence in the law of large numbers and the consistency of nonparametric regression when the errors are martingale differences are given.

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.