Almost Sure Convergence

This paper introduces a new stochastic process, a collection of U-statis-tics indexed by a family of symmetric kemels. Conditions are found for the uniform almost-sure convergence of a sequence of such processes. Rates of convergence are obtained. An application to cross-validation ...

Nous établissons un majorant de la vitesse de convergence presque sûre de l'estimateur à noyau du mode, en utilisant des résultats de type loi du logarithme itéré.We obtain rate of almost sure convergence of kernel estimators of the mode, using law of the iterated logarithm type results

In (Ordóñez Cabrera and Volodin, J. Math. Anal. Appl. 305:644–658, 2005), the authors introduce the notion of h-integrability of an array of random variables with respect to an array of constants, and obtained some mean convergence theorems for weighted sums of random variables subject to some special kinds of dependence. In view of the important role played by conditioning and dependence in the models used to describe many situations in the applied sciences, the concepts and results in the aforementioned paper are extended herein to the case of randomly weighted sums of dependent random variables when a sequence of conditioning sigma-algebras is given. The dependence conditions imposed on the random variables (conditional negative quadrant dependence and conditional strong mixing) as well as the convergence results obtained are conditional relative to the conditioning sequence of sigma-algebras. In the last section, a strong conditional convergence theorem is also established by us...

A probabilistic normed space (PN space) is a natural generalization of an ordinary normed linear space. In PN space, the norms of the vectors are represented by prob-ability distribution functions rather than a positive number. Such spaces were first introduced by AN Šerstnev in 1963 ...

We consider a class of stochastic mathematical programs with complementarity constraints, in which both the objective and the constraints involve limit functions or expectations that need to be estimated or approximated. Such programs can be used for modeling \average" or steady-state behavior of complex stochastic systems. Recently, simulation-based methods have been successfully used for solving challenging stochastic optimization problems and

Using the explicit representations of the Brownian motions on the hyperbolic spaces, we show that their almost sure convergence and the central limit theorems for the radial components as time tends to infinity are easily obtained. We also give a straightforward strategy to obtain the explicit expressions for the limit distributions or the Poisson kernels.

We consider a random variable X satisfying almost-sure conditions involving G: = ˙ DX, −DL −1 X ¸ where DX is X’s Malliavin derivative and L −1 is the inverse Ornstein-Uhlenbeck operator. A lower-(resp. upper-) bound condition on G is proved to imply a Gaussian-type lower (resp. upper) bound on the tail P [X> z]. Bounds of other natures are also given. A key ingredient is the use of Stein’s lemma, including the explicit form of the solution of Stein’s equation relative to the function 1x>z, and its relation to G. Another set of comparable results is established, without the use of Stein’s lemma, using instead a formula for the density of a random variable based on G, recently devised by the author and Ivan Nourdin. As an application, via a Mehler-type formula for G, we show that the Brownian polymer in a Gaussian environment which is white-noise in time and positively correlated in space has deviations of Gaussian type and a fluctuation exponent χ = 1/2. We also show this expo...

Abstract-We consider the problem of reconstructing a de-terministic data field from binary quantized noisy observations of sensors randomly deployed over the field domain. Our focus is on the extremes of lack of control in the sensor deployment, arbitrariness and lack of knowledge of the ...

In this work, we study the optimal discretization error of stochastic integrals, in the context of the hedging error in a multidimensional It\^{o} model when the discrete rebalancing dates are stopping times. We investigate the convergence, in an almost sure sense, of the renormalized quadratic variation of the hedging error, for which we exhibit an asymptotic lower bound for a large class of stopping time strategies. Moreover, we make explicit a strategy which asymptotically attains this lower bound a.s. Remarkably, the results hold under great generality on the payoff and the model. Our analysis relies on new results enabling us to control a.s. processes, stochastic integrals and related increments.

In this article, a new framework for evolutionary algorithms for approximating the efficient set of a multiobjective optimization (MOO) problem with continuous variables is presented. The algorithm is based on populations of variable size and exploits new elite preserving rules for selecting alternatives generated by mutation and recombination. Together with additional assumptions on the considered MOO problem and further specifications on the algorithm, theoretical results on the approximation quality such as convergence in probability and almost sure convergence are derived.

We investigate sample average approximation of a general class of onestage stochastic mathematical programs with equilibrium constraints. By using graphical convergence of unbounded set-valued mappings, we demonstrate almost sure convergence of a sequence of stationary points of sample average approximation problems to their true counterparts as the sample size increases. In particular we show the convergence of M(Mordukhovich)-stationary point and C(Clarke)-stationary point of the sample average approximation problem to those of the true problem. The research complements the existing work in the literature by considering a general equilibrium constraint to be represented by a stochastic generalized equation and exploiting graphical convergence of coderivative mappings.

By applying a recent result of Hu et al. [Stochastic Anal. Appl., 17 (1999), pp. 963--992], we extend and generalize the complete convergence results of Pruitt [ J. Math. Mech., 15 (1966), pp. 769--776] and Rohatgi [Proc. Cambridge Philos. Soc., 69 (1971), pp. 305--307] to arrays of row-wise independent Banach space valued random elements. No assumptions are made concerning the geometry of the underlying Banach space. Illustrative examples are provided comparing the various results.