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### A natural derivative on [0, n] and a binomial Poincaré inequality

ESAIM: Probability and Statistics

We consider probability measures supported on a finite discrete interval [0, n]. We introduce a new finite difference operator ∇n, defined as a linear combination of left and right finite differences. We show that this operator ∇n plays a key role in a new Poincaré (spectral gap) inequality with respect to binomial weights, with the orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator. We briefly discuss the relationship of this operator to the problem of optimal transport...

### A note on the application of integrals involving cyclic products of kernels.

Qüestiió

In statistics of stochastic processes and random fields, a moment function or a cumulant of an estimate of either the correlation function or the spectral function can often contain an integral involving a cyclic product of kernels. We define and study this class of integrals and prove a Young-Hölder inequality. This inequality further enables us to study asymptotics of the above mentioned integrals in the situation where the kernels depend on a parameter. An application to the problem of estimation...

### Alternative models in precipitation analysis.

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

### An extension theorem to rough paths

Annales de l'I.H.P. Analyse non linéaire

### Asymptotic rate of convergence in the degenerate U-statistics of second order

Banach Center Publications

Let X,X₁,...,Xₙ be independent identically distributed random variables taking values in a measurable space (Θ,ℜ ). Let h(x,y) and g(x) be real valued measurable functions of the arguments x,y ∈ Θ and let h(x,y) be symmetric. We consider U-statistics of the type $T\left(X₁,...,Xₙ\right)={n}^{-1}{\sum }_{1\le iLet{q}_{i}\left(i\ge 1\right)beeigenvaluesoftheHilbert-Schmidtoperatorassociatedwiththekernelh\left(x,y\right),andq₁bethelargestinabsolutevalueone.Weprovethat}$Δn = ρ(T(X₁,...,Xₙ),T(G₁,..., Gₙ)) ≤ (cβ’1/6)/(√(|q₁|) n1/12)$,$where ${G}_{i}$, 1 ≤ i ≤ n, are i.i.d. Gaussian random vectors, ρ is the Kolmogorov (or uniform) distance and ${\beta }^{\text{'}}:=E|h\left(X,X₁\right)|³+{E|h\left(X,X₁\right)|}^{18/5}+E|g\left(X\right)|³+{E|g\left(X\right)|}^{18/5}+1<\infty$.

### Bernstein inequality for the parameter of the pth order autoregressive process AR(p)

Applicationes Mathematicae

The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter θ̃ of the first-order autoregressive process taking values in a real separable Banach space B (ARB(1)), if it satisfies the following relation: $X{̃}_{t}=\theta ̃X{̃}_{t-1}+\epsilon {̃}_{t}$. In this paper we study the convergence in distribution of the linear operator $I\left(\theta {̃}_{T},\theta ̃\right)=\left(\theta {̃}_{T}-\theta ̃\right)\theta {̃}^{T-2}$ for ||θ̃|| > 1 and so we construct inequalities of Bernstein type for this operator.

### Causality and stochastic realization.

International Journal of Mathematics and Mathematical Sciences

### Conditional Banach Spaces, Conditional Projections and Generalized Martingales.

Mathematica Scandinavica

### Entropy estimate for $k$-monotone functions via small ball probability of integrated Brownian motions.

Electronic Communications in Probability [electronic only]

### Estimation and tests of the discrete probability law based on the empirical generating function, (two dimensional case).

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales

### Gibbsian fields associated to exponentially decreasing quadratic potentials

Annales de l'I.H.P. Probabilités et statistiques

### Integral criteria for transportation-cost inequalities.

Electronic Communications in Probability [electronic only]

### Large deviations for independent random variables – Application to Erdös-Renyi’s functional law of large numbers

ESAIM: Probability and Statistics

A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum }_{1}^{n}𝐟\left({x}_{i}^{n}\right)·{Z}_{i}^{n}$ where the deterministic probability measure $\frac{1}{n}{\sum }_{1}^{n}{\delta }_{{x}_{i}^{n}}$ converges weakly to a probability measure $R$ and ${\left({Z}_{i}^{n}\right)}_{i\in ℕ}$ are ${ℝ}^{d}$-valued independent random variables whose distribution depends on ${x}_{i}^{n}$ and satisfies the following exponential moments condition:$\phantom{\rule{-56.9055pt}{0ex}}\underset{i,n}{sup}𝔼{\mathrm{e}}^{{\alpha }^{*}|{Z}_{i}^{n}|}<+\infty \phantom{\rule{1.0em}{0ex}}\mathrm{forsome}\phantom{\rule{1.0em}{0ex}}0<{\alpha }^{*}<+\infty .$In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result, we extend...

### Large deviations for independent random variables – Application to Erdös-Renyi's functional law of large numbers

ESAIM: Probability and Statistics

A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum }_{1}^{n}𝐟\left({x}_{i}^{n}\right)·{Z}_{i}^{n}$ where the deterministic probability measure $\frac{1}{n}{\sum }_{1}^{n}{\delta }_{{x}_{i}^{n}}$ converges weakly to a probability measure R and ${\left({Z}_{i}^{n}\right)}_{i\in ℕ}$ are ${ℝ}^{d}$-valued independent random variables whose distribution depends on ${x}_{i}^{n}$ and satisfies the following exponential moments condition: $\underset{i,n}{sup}𝔼{\mathrm{e}}^{{\alpha }^{*}|{Z}_{i}^{n}|}<+\infty \phantom{\rule{1.0em}{0ex}}\mathrm{forsome}\phantom{\rule{1.0em}{0ex}}0<{\alpha }^{*}<+\infty .$ In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result,...

### Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies

ESAIM: Probability and Statistics

### Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies

ESAIM: Probability and Statistics

We observe an infinitely dimensional Gaussian random vector x = ξ + v where ξ is a sequence of standard Gaussian variables and v ∈ l2 is an unknown mean. We consider the hypothesis testing problem H0 : v = 0versus alternatives ${H}_{\epsilon ,\tau }:v\in {V}_{\epsilon }$ for the sets ${V}_{\epsilon }={V}_{\epsilon }\left(\tau ,{\rho }_{\epsilon }\right)\subset {l}_{2}$. The sets Vε are lq-ellipsoids of semi-axes ai = i-s R/ε with lp-ellipsoid of semi-axes bi = i-r pε/ε removed or similar Besov bodies Bq,t;s (R/ε) with Besov bodies Bp,h;r (pε/ε) removed. Here $\tau =\left(\kappa ,R\right)$ or $\tau =\left(\kappa ,h,t,R\right);\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\kappa =\left(p,q,r,s\right)$ are the parameters which define the sets Vε for given radii...

### Model selection for regression on a random design

ESAIM: Probability and Statistics

We consider the problem of estimating an unknown regression function when the design is random with values in ${ℝ}^{k}$. Our estimation procedure is based on model selection and does not rely on any prior information on the target function. We start with a collection of linear functional spaces and build, on a data selected space among this collection, the least-squares estimator. We study the performance of an estimator which is obtained by modifying this least-squares estimator on a set of small probability....

### Model selection for regression on a random design

ESAIM: Probability and Statistics

We consider the problem of estimating an unknown regression function when the design is random with values in ${ℝ}^{k}$. Our estimation procedure is based on model selection and does not rely on any prior information on the target function. We start with a collection of linear functional spaces and build, on a data selected space among this collection, the least-squares estimator. We study the performance of an estimator which is obtained by modifying this least-squares estimator on a set of small...

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