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We aim to obtain very tight exponential bounds for hyperbolic tangent by first establishing simple algebraic bounds for this function. Our inequalities refine a double inequality for tanh x recently proved by Zhang and Chen. In addition, a graphical analysis is carried out, and a number of auxiliary lemmas may be of use on their own.

In this note, a new strategy is proposed to obtain bounds for functions having product decompositions. Applications are given for trigonometric and hyperbolic functions, thus improving some existing inequalities in the literature. Some graphics illustrate the findings.

Nous considerons le probleme inverse stochastique de reconstruire $f$ a partir de donnees bruitees $Y=Tf+\sigma \xi $ ou $\xi $ est un bruite blanc et $T$ un operateur compact entre espaces de Hilbert. Considerant des methodes de reconstruction generales de la forme $\hat{f}_{\alpha }=q_{\alpha }(T^{*}T)T^{*}Y$ avec un filtre ordonne $q_{\alpha }$, nous examinons le choix du parametre de regularisation $\alpha $ en minimisant un estimateur non biaise du risque predictif $\mathbb{E}[\Vert Tf-T\hat{f}_{\alpha }\Vert^{2}]$. Le parametre correspondant $\alpha_{\mathrm{pred}}$ et son utilisation sont bien connus dans la litterature mais les inegalites oracles et les resultats d’optimalite dans ce cadre general sont inconnus. Nous prouvons une inegalite oracle (generalisee), qui relie le risque direct $\mathbb{E}[\Vert f-\hat{f}_{\alpha_{\mathrm{pred}}}\Vert^{2}]$ au risque de l’oracle de prediction $\inf_{\alpha >0}\mathbb{E}[\Vert Tf-T\hat{f}_{\alpha }\Vert^{2}]$. A partir de cette inegalite oracle nous sommes alors capable de conclure que la regle de choix du parametre examine est d’ordre optimale au sens minimax. Finalement nous presentons aussi des simulations numeriques qui confirment l’optimalite de l’ordre de la methode et la qualite du choix du parametre dans des situations discretes.

This paper is devoted to obtain sharp exponential and polynomial upper bounds for the ratio functions cosh x/cos x and sinh x/sin x. The proofs are based on the use of a refinement of a special case of the Bernoulli inequality and infinite products.

In this note, we improve the useful inequalities sin(x)sinh(x) < x^2 and cos(x)cosh(x) < 1 by the use of infinite products.

We consider the class of Hilbert valued autoregressive process with random coefficients (RCARH). For this class we dérivé limit theorems : strong law of great numbers, central limit theorem, compact law of the iterated logarithm and exponential inequalities and we also obtain rates of convergence. These results are crucial in the frarnework of Hilbert autoregressive processes statistical analysis. Our approach is based on a suitable décomposition of Hilbert valued martingale differences and martingale approximation.

The thesis focuses on the superconcentration phenomenon which appears in the study of the fluctuations of various moelds from current research (random matrices, spin glasses, discrete Gaussien free field, percolation,...). More precisely, the thesis mainly deals with superconcentration inequalities at an exponentiel level ; in particular for supremum of familu of Gaussian random variables. The principal tools used during this study are the hypercontractive property satisfied by some Markov semi-groups ; this approach leads to an extension of higher order of an inequality due to M. Talagrand. The first part of the thesis exposes the fundamental notions of concentration of measure, interpolation methods with Markovians semi-groups, functional inequalities, optimal transport and isoperimetry. Then, a survey of the literature concerning superconcentration phenomenon is done. The second part of the manuscript bring together, in different chapters, the results obtained during the thesis. Most of them are based on the dynamical representation of the variance along the semi-group of Ornstein-Uhlenbeck and its hypercontractive property. New ineqaulities are obtained at an exponential level and are illustrated on examples coming from extreme theory. This hypercontractive framework also gave birth to a new inequality on the discrete cube which leads to an application on the influence of second order of boolean functions. Finally, the last chapter is about the superconcentration phenomenon with an optimal transport approach. Some non asymptotic bounds on the variance and deviations inequalities are obtained for the maximum of an i.i.d. sample. Again, illustrations for usual laws of probability, belonging to different domain of attraction from extreme theory, are given.; Cette thèse porte sur le phénomène de superconcentration qui apparaît dans l'étude des fluctuations de divers modèles de la recherche actuelle (matrices aléatoires, verres de spins, champ libre gaussien discret, percolation,...). Plus particulièrement, la thèse est consacrée à l'examen d'inégalités de superconcentration à l'échelle exponentielle ; notamment pour des supremum de familles gaussiennes. Les outils mis en œuvre comprennent la propriété d'hypercontractivité de semi-groupes de Markov. Par ailleurs, celle-ci a conduit à une version d'ordre supérieur d'une inégalité sur la variance de M. Talagrand. La première partie de la thèse présente brièvement les notions essentielles de la théorie classique de la concentration de la mesure ainsi que les principaux outils, à savoir : méthodes d'interpolations à l'aide de semi-groupes markoviens, inégalités fonctionnelles, transport optimal et isopérimétrie. Un survol de la littérature existante est ensuite proposé. La deuxième partie du manuscrit rassemble, dans différents chapitres, les travaux que nous avons effectués durant cette thèse. Une grande partie de ceux-ci repose sur la représentation dynamique de la variance le long du semi-groupe d'Ornstein-Uhlenbeck et sa propriété d'hypercontractivité. De nouvelles inégalités de superconcentration sont obtenues au niveau exponentiel et illustrées sur des exemples provenant de la théorie des extrêmes. Le cadre de l'hypercontractivité a également conduit à une nouvelle inégalité sur le cube discret, celle-ci permettant une application sur l'influence d'ordre deux de fonctions booléennes. Enfin, le dernier chapitre aborde la phénomène de superconcentration par le transport optimal. Des majorations de la variance et des inégalités de déviations non asymptotiques pour le maximum de variables aléatoires indépendantes et de même loi sont obtenues. A nouveau, des illustrations pour des lois usuelles, appartenant aux différents domaines d'attraction de la théorie des extrêmes, sont proposées

The thesis focuses on the superconcentration phenomenon which appears in the study of the fluctuations of various moelds from current research (random matrices, spin glasses, discrete Gaussien free field, percolation,...). More precisely, the thesis mainly deals with superconcentration inequalities at an exponentiel level ; in particular for supremum of familu of Gaussian random variables. The principal tools used during this study are the hypercontractive property satisfied by some Markov semi-groups ; this approach leads to an extension of higher order of an inequality due to M. Talagrand. The first part of the thesis exposes the fundamental notions of concentration of measure, interpolation methods with Markovians semi-groups, functional inequalities, optimal transport and isoperimetry. Then, a survey of the literature concerning superconcentration phenomenon is done. The second part of the manuscript bring together, in different chapters, the results obtained during the thesis. Most of them are based on the dynamical representation of the variance along the semi-group of Ornstein-Uhlenbeck and its hypercontractive property. New ineqaulities are obtained at an exponential level and are illustrated on examples coming from extreme theory. This hypercontractive framework also gave birth to a new inequality on the discrete cube which leads to an application on the influence of second order of boolean functions. Finally, the last chapter is about the superconcentration phenomenon with an optimal transport approach. Some non asymptotic bounds on the variance and deviations inequalities are obtained for the maximum of an i.i.d. sample. Again, illustrations for usual laws of probability, belonging to different domain of attraction from extreme theory, are given.; Cette thèse porte sur le phénomène de superconcentration qui apparaît dans l'étude des fluctuations de divers modèles de la recherche actuelle (matrices aléatoires, verres de spins, champ libre gaussien discret, percolation,...). Plus particulièrement, la thèse est consacrée à l'examen d'inégalités de superconcentration à l'échelle exponentielle ; notamment pour des supremum de familles gaussiennes. Les outils mis en œuvre comprennent la propriété d'hypercontractivité de semi-groupes de Markov. Par ailleurs, celle-ci a conduit à une version d'ordre supérieur d'une inégalité sur la variance de M. Talagrand. La première partie de la thèse présente brièvement les notions essentielles de la théorie classique de la concentration de la mesure ainsi que les principaux outils, à savoir : méthodes d'interpolations à l'aide de semi-groupes markoviens, inégalités fonctionnelles, transport optimal et isopérimétrie. Un survol de la littérature existante est ensuite proposé. La deuxième partie du manuscrit rassemble, dans différents chapitres, les travaux que nous avons effectués durant cette thèse. Une grande partie de ceux-ci repose sur la représentation dynamique de la variance le long du semi-groupe d'Ornstein-Uhlenbeck et sa propriété d'hypercontractivité. De nouvelles inégalités de superconcentration sont obtenues au niveau exponentiel et illustrées sur des exemples provenant de la théorie des extrêmes. Le cadre de l'hypercontractivité a également conduit à une nouvelle inégalité sur le cube discret, celle-ci permettant une application sur l'influence d'ordre deux de fonctions booléennes. Enfin, le dernier chapitre aborde la phénomène de superconcentration par le transport optimal. Des majorations de la variance et des inégalités de déviations non asymptotiques pour le maximum de variables aléatoires indépendantes et de même loi sont obtenues. A nouveau, des illustrations pour des lois usuelles, appartenant aux différents domaines d'attraction de la théorie des extrêmes, sont proposées

Methods are proposed to construct empirical measures when there are missing terms among the components of a random vector. Fur- thermore, Vapnik-Chevonenkis type exponential bounds are obtained on the uniform deviations of these estimators, from the true probabilities. These results can then be used to deal with classical problems such as statistical classification, via empirical risk minimization, when there are missing covariates among the data. Another application involves the uni- form estimation of a distribution function. AMS 2000 subject classifications: Primary 60G50, 62G15; secondary 62H30.

Revised version; In a non-parametric framework, we establish some non-asymptotic bounds for self-normalized sums and quadratic forms in the multivariate case for symmetric and general random variables. This bounds are entirely explicit and essentially depends in the general case on the kurtosis of the Euclidean norm of the standardized random variables.

In this paper we develop a framework for computing upper and lower bounds of an exponential form for a class of single server queueing systems with non-renewal inputs. These bounds generalize Kingman's bounds for queues with renewal inputs.

We consider a storage process with finite or infinite capacity having a compound Poisson process as input and general release rule. For this process we derive some exponential type upper and lower bounds for hitting time distributions by means of martingale theory.

In this paper we study GI/G/I queueing systems with finite capacity. Our main objective is to find exponential bounds for the limiting probability that an arriving customer causes an excess of the system capacity. The derivation of these bounds is based on a comparison of the finite and infinite capacity systems.

This paper presents some generalizations of S. N. Bernstein's exponential bounds on probabilities of large deviations to the vector case. Inequalities for probabilities of large deviations of sums of independent random vectors are derived under a Cramer's type restriction on the rate of growth of absolute moments of the summands. Estimates are obtained for random vectors with values in Banach space, Sharper bounds hold in the case of finite-dimensional Euclidean or separable Hilbert spaces.

The uniform empirical process $U_n$ is considered as a process indexed by intervals. Powerful new exponential bounds are established for the process indexed by both "points" and intervals. These bounds trivialize the proof of the Chibisov-O'Reilly theorem concerning the convergence of the process with respect to $\|\cdot/q\|$-metrics and are used to prove an interval analogue of the Chibisov-O'Reilly theorem. A strong limit theorem related to the well-known Holder condition for Brownian bridge $U$ is also proved. Connections with related work of Csaki, Eicker, Jaeschke, and Stute are mentioned. As an application we introduce a new statistic for testing uniformity which is the natural interval analogue of the classical Anderson-Darling statistic.

Motivated by the problem of establishing laws of the iterated logarithm for least squares estimates in regression models and for partial sums of linear processes, we prove a general $\log \log$ law for weighted sums of the form $\sum^\infty_{i=-\infty} a_{ni}\varepsilon_i$, where the $\varepsilon_i$ are independent random variables with zero means and a common variance $\sigma^2$, and $\{a_{ni}: n \geq 1, -\infty < i < \infty\}$ is a double array of constants such that $\sum^\infty_{i=-\infty} a^2_{ni} < \infty$ for every $n$. Besides applying the general theorem to least squares estimates and linear processes, we also use it to improve earlier results in the literature concerning weighted sums of the form $\sum^n_{i=1} f(i/n)\varepsilon_i$.

The simple hypothesis is tested that the distribution of independent random variables $X_1, X_2, \cdots, X_n$ is a given probability measure $P_0$. Let $\pi_n$ be any sequence of partitions. The alternative hypothesis is the set of probability measures $P$ with $\sum_{A\in \pi_n}|P(A) - P_0(A)| \geq \delta$, where $\delta > 0$. Note the dependence of this set of alternatives on the sample size. It is shown that if the effective cardinality of the partitions is of the same order as the sample size, then sequences of tests exist with uniformly exponentially small probabilities of error. Conversely, if the effective cardinality is of larger order than the sample size, then no such sequence of tests exists. The effective cardinality is the number of sets in the partition which exhaust all but a negligible portion of the probability under the null hypothesis.

Let $\{X_i\}$ be a sequence of independent, identically distributed nondegenerate random variables and $S_n = \sum^n_{i = 1}X_i$. We consider the question for various centering sequences $\{\alpha_n\}$: when is it possible to find a positive, monotone sequence $\{\beta_n\}$ such that $\lim \sup \beta^{-1}_n (S_n - \alpha_n) = c$ a.s., $c$ a finite nonzero constant? If $\alpha_n = \operatorname{med} S_n$, we obtain a necessary and sufficient condition for this. An important corollary is a one-sided version of the Hartman-Wintner law of the iterated logarithm: if $E(X^+)^2 < \infty$, then it is always possible to find such a norming sequence. Explicit norming sequences are given which are easy to obtain. Necessary and sufficient conditions are also given for being able to find a norming sequence $\{\beta_n\}$ for the two-sided problem $(\lim \sup \beta^{-1}_n |S_n - \alpha_n| = c$ a.s.) when $\alpha_n = ES_n$ and $\alpha_n = 0$. The two-sided problem with $\alpha_n = \operatorname{med} S_n$ was solved by Kesten. The one-sided problem remains open for $\alpha_n = ES_n$ and $\alpha_n = 0$. Examples are given which illustrate the advantage of considering different centering sequences. A one-sided version of Strassen's converse to the law of the iterated logarithm is also given: if $\lim \sup S_n/ \sqrt{2n \log \log n} = 1$ a.s., then $EX = 0, EX^2 = 1$.

Let $X_1, X_2,\cdots$ be a sequence of independent random variables, each with zero mean and finite variance. Define $S_n = \sum^n_{i=1} X_i, s_n^2 = E(S_n^2), t_n^2 = 2 \log \log s_n^2$ and $\Lambda = \lim \sup_{n\rightarrow\infty} S_n/(s_n t_n)$. Suppose that $|X_n| \leqq c_n s_n$ a.s. for all $n$ and some real sequence $\{c_n\}$ and that $s_n \rightarrow \infty$, and let $\nu = \lim \sup_{n\rightarrow\infty} t_nc_n$. By Kolmogorov's law of the iterated logarithm, $\Lambda = 1$ if $\nu = 0$. Egorov proved that $0 \leqq \Lambda < \infty$, in every case. In this paper it will be shown that, if $\nu < \infty$, then $0 < \Lambda \leqq 1 + \sum^\infty_{j=3} \nu^{j-2}/j!$. A similar result for certain classes of unbounded random variables will also be presented.

In this paper, we study weighted sums $\sum^n_{i=1} c_{n-i} X_i$ of i.i.d. zero-mean random variables $X_1, X_2, \cdots$, under the condition that the sequence $(c_n)$ is square summable. It is proved that such weighted sums are, with probability 1, of smaller order than $n^{1/\alpha}$ (respectively $\log n$, etc.) $\operatorname{iff} E|X_1|^\alpha < \infty$ (respectively $Ee^{t|X_1|} < \infty$ for all $t < \infty$, etc.). Certain analogs of the law of the iterated logarithm for such weighted sums are also obtained.

In this note, new sharp bounds for trigonometric functions are proved. We provide alternative proofs to existing results on exponential bounds, with some improvements, by using infinite products and the so-called Bernoulli inequality.

In this note, natural exponential bounds for coshx are established. The inequalities thus obtained are interesting and sharp.