An invariance principle for lattices of dependent random variables

Summary It is shown that for -mixing arrays of Banach space valued random vectors the central limit theorem implies the invariance principle. Applying this result to lattices of random variables a higher dimensional invariance principle under dependence assumptions is obtained.Dedicated to Professor Leopold Schmetterer     

An almost sure invariance principle for triangular arrays of banach space valued random variables

Summary We give a simpler proof of the probability invariance principle for triangular arrays of independent identically distributed random variables with values in a separable Banach space, recently proved by de Acosta [1], and improve this result to an almost sure invariance principle.     

Invariance principles for the law of the iterated logarithm under <Emphasis Type="Italic">G</Emphasis>-framework

We obtain a general invariance principle of G-Brownian motion for the law of the iterated logarithm (LIL for short). For continuous bounded independent and identically distributed random variables in G-expectation space, we also give an invariance principle for LIL. In some sense, this result is an extension of the classical Strassen’s invariance principle to the case where probability measure is no longer additive. Furthermore, we give some examples as applications.     

Strong Laws for L p -Norms of Empirical and Related Processes

We obtain lim sup and lim inf results for the L _p -norms (1 ≤ p < ∞) of empirical and quantile processes. We prove these results combining theorems for sums of Banach space valued random variables with invariance principles.     

Laws of the iterated logarithm and an almost sure invariance principle for mixing <Emphasis Type="Italic">B</Emphasis>-valued random variables and autoregressive processes

Under optimal moment conditions, we prove the compact law of the iterated logarithm and the almost sure invariance principle for ψ-mixing random variables with values in type 2 Banach spaces. These results, together with the bounded law of the iterated logarithm proved earlier by author, allow us to prove the same kind of results for the Banach space valued autoregressive processes with ψ-mixing innovations. The results for autoregressive processes can be considered as asymptotic properties of the estimator of mean.     

Weighted Invariance Principle for Banach Space-Valued Random Variables

A weighted weak invariance principle for nonseparable Banach space-valued functions is described via asymptotic behavior of a weighted Wiener process. It is proved that, unlike the usual weak invariance principle, the weighted variant cannot be characterized via validity of a central limit theorem in a Banach space. A strong invariance principle is introduced in the present context and used to prove the weighted weak invariance principle that we seek herewith. The result then is applied to empirical processes.     

Invariance principles for the law of the iterated logarithm under G-framework

We obtain a general invariance principle of G-Brownian motion for the law of the iterated logarithm(LIL for short). For continuous bounded independent and identically distributed random variables in G-expectation space, we also give an invariance principle for LIL. In some sense, this result is an extension of the classical Strassen's invariance principle to the case where probability measure is no longer additive. Furthermore,we give some examples as applications.     

The Probabilistic Feynman–Kac Formula for an Infinite-Dimensional Schrödinger Equation with Exponential and Singular Potentials

Using the probabilistic Feynman–Kac formula, the existence of solutions of the Schrödinger equation on an infinite dimensional space E is proven. This theorem is valid for a large class of potentials with exponential growth at infinity as well as for singular potentials. The solution of the Schrödinger equation is local with respect to time and space variables. The space E can be a Hilbert space or other more general infinite dimensional spaces, like Banach and locally convex spaces (continuous functions, test functions, distributions). The specific choice of the infinite dimensional space corresponds to the smoothness of the fields to which the Schrödinger equation refers. The results also express an infinite-dimensional Heisenberg uncertainty principle: increasing of the field smoothness implies increasing of divergence of the momentum part of the quantum field Hamiltonian.     

A general estimate in the invariance principle

We obtain estimates for the accuracy with which a random broken line constructed from sums of independent nonidentically distributed random variables can be approximated by a Wiener process. All estimates depend explicitly on the moments of the random variables; meanwhile, these moments can be of a rather general form. In the case of identically distributed random variables we succeed for the first time in constructing an estimate depending explicitly on the common distribution of the summands and directly implying all results of the famous articles by Komlós, Major, and Tusnády which are devoted to estimates in the invariance principle.     

On the invariance principle for stationary ϕ-mixing triangular arrays with infinitely divisible limits

Summary Given a stationary, -mixing triangular array of Banach space valued random vectors whose row sums converge weakly to an infinitely divisible probability measure, necessary and sufficient conditions for the validity of the corresponding invariance principle in distribution are given.     

On the S-transform over a Banach algebra

The S-transform is shown to satisfy a specific twisted multiplicativity property for free random variables in a B-valued Banach noncommutative probability space, for an arbitrary unital complex Banach algebra B. Also, a new proof of the additivity of the R-transform in this setting is given.     

Invariant subspaces of finite codimension in Banach spaces of analytic functions

We give a characterization of invariant subspaces of finite codimension in Banach spaces of vector-valued analytic functions in several variables, where invariant refers to invariance under multiplication by any polynomial. We obtain very weak conditions under which our characterization applies, that unifies and improves a number of previous results. In the vector-valued case, the results are new even for one complex variable. As a concrete application in several variables, we consider the Bergman space on a strictly pseudo-convex domain, and we improve previous results (assuming C^∞-boundary) to the case of C²-boundary.     

Statistiques d'une saisonnalité perturbée par un processus a représentation autorégressiveStatistics of seasonality perturbed by time continuous processes with autoregressive representation

We consider limit theorems for an estimator of a seasonality when it is perturbed by a time continuous process admitting a Banach autoregressive representation. From the compact iterated logarithm law we derive confidence regions for a(·) in the Banach space of continuous functions. When a(·) belongs to a finite dimensional subspace, we study the estimation of a(·) by projection and we estimate the dimension when it is unknown. To cite this article: T. Mourid, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 909–912.     

Weak invariance principle in some Besov spaces for stationary martingale differences

We extend the classical Donsker weak invariance principle to some Besov space framework. We consider polygonal line processes built from partial sums of stationary martingale differences and of independent and identically distributed random variables. The results obtained are shown to be optimal.     

On the diffusive behavior of isotropic diffusions in a random environment

We present here results concerning the asymptotic behavior of isotropic diffusions in random environment that are small perturbations of Brownian motion. When the space dimension is three or more we prove an invariance principle as well as transience. Our methods also apply to questions of homogenization in random media. To cite this article: A.-S. Sznitman, O. Zeitouni, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A large deviation principle for bootstrapped sample means

A large deviation principle for bootstrapped sample means is established. It relies on the Bolthausen large deviation principle for sums of i.i.d. Banach space valued random variables. The rate function of the large deviation principle for bootstrapped sample means is the same as the classical one.

     

The limit theorems for random walk with state space ℝ in a space-time random environment

We consider a discrete time random environment. We state that when the random walk on real number space in a environment is i.i.d., under the law, the law of large numbers, iterated law and CLT of the process are correct space-time random marginal annealed Using a martingale approach, we also state an a.s. invariance principle for random walks in general random environment whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain.     

Existence of the linear prediction for Banach space valued Gaussian processes

The correspondence between Gaussian stochastic processes with values in a Banach space E and cylindrical processes which are related to them is studied. It is shown that the linear prediction of an E-valued Gaussian process is an E-valued random variable as well as the spectral measure of an E-valued Gaussian stationary process is a Gaussian random measure.     

Rosenthal inequalities in noncommutative symmetric spaces

We give a direct proof of the ‘upper’ Khintchine inequality for a noncommutative symmetric (quasi-)Banach function space with nontrivial upper Boyd index. This settles an open question of C. Le Merdy and the fourth named author (Le Merdy and Sukochev, 2008 [24]). We apply this result to derive a version of Rosenthal?s theorem for sums of independent random variables in a noncommutative symmetric space. As a result we obtain a new proof of Rosenthal?s theorem for (Haagerup) Lp-spaces.     

An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables

Summary In this paper we establish an almost sure invariance principle with an error termo((t log logt)^1/2) (ast) for partial sums of stationary ergodic martingale difference sequences taking values in a real separable Banach space. As partial sums of weakly dependent random variables can often be well approximated by martingales, this result also leads to almost sure invariance principles for a wide class of stationary ergodic sequences such as ø-mixing and -mixing sequences and functionals of such sequences. Compared with previous related work for vector valued random variables (starting with an article by Kuelbs and Philipp [27]), the present approach leads to a unification of the theory (at least for stationary sequences), moment conditions required by earlier authors are relaxed (only second order weak moments are needed), and our proofs are easier in that we do not employ estimates of the rate of convergence in the central limit theorem but merely the central limit theorem itself.