Almost sure critical paths

M. Kress proved for a special case of Location-Scale probability distributions there always exists a probability level for which the Chance Constrained Critical Path (CCCP) remains unchanged for all probabilities greater than or equal to that value. His chance constrained problem has zero-order decision rules and individual chance constraints. This paper extends his results to most of the common probability distributions.     

Almost sure behaviour ofF-valued random fields

Summary LetX _n, n ^d be a field of independent random variables taking values in a semi-normed measurable vector spaceF. For a broad class of fields _n, ^d of positive numbers, the almost sure behaviour of _knX_k/_n, n ^d is studied. The main result allows us to deduce some new and well-known theorems for fields of independentF random variables from related results for fields of independent real random variables.Supported in part by the Youth Science Foundation of China, No. 19001018Supported by the National Natural Science Foundation of China     

Almost sure polynomial asymptotic stability of stochastic difference equations

In this paper, we establish the almost sure asymptotic stability and decay results for solutions of an autonomous scalar difference equation with a nonhyperbolic equilibrium at the origin, which is perturbed by a random term with a fading state-independent intensity. In particular, we show that if the unbounded noise has tails that fade more quickly than polynomially, then the state-independent perturbation dies away at a sufficiently fast polynomial rate in time, and if the autonomous difference equation has a polynomial nonlinearity at the origin, then the almost sure polynomial rate of decay of solutions can be determined exactly. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 17, Differential and Functional Differential Equations. Part 3, 2006.     

Almost sure exponential behavior of a directed polymer in a fractional Brownian environment

This paper studies the asymptotic behavior of a one-dimensional directed polymer in a random medium. The latter is represented by a Gaussian field BH on R₊×R with fractional Brownian behavior in time (Hurst parameter H) and arbitrary function-valued behavior in space. The partition function of such a polymer is

     

Almost sure limiting behaviour of first crossing points of Gaussian sequences

Let {K_k,k∈Z} be a stationary, normalized Gaussian sequence and define τ_β=min(k:X_k>?βk} the first crossing point of the Gaussian sequence with the moving boundary ?βt. For β→0 we discuss in this paper the a.s. stability, the a.s. relative ability of τ_β and an iterated logarithm law for τ_β, depending on the correlation function.     

Almost sure exponential behaviour for a parabolic SPDE on a manifold

We derive an upper bound on the large-time exponential behavior of the solution to a stochastic partial differential equation on a compact manifold with multiplicative noise potential. The potential is a random field that is white-noise in time, and Hölder-continuous in space. The stochastic PDE is interpreted in its evolution (semigroup) sense. A Feynman–Kac formula is derived for the solution, which is an expectation of an exponential functional of Brownian paths on the manifold. The main analytic technique is to discretize the Brownian paths, replacing them by piecewise-constant paths. The error committed by this replacement is controlled using Gaussian regularity estimates; these are also invoked to calculate the exponential rate of increase for the discretized Feynman–Kac formula. The error is proved to be negligible if the diffusion coefficient in the stochastic PDE is small enough. The main result extends a bound of Carmona and Viens (Stochast. Stochast. Rep. 62 (3–4) (1998) 251) beyond flat space to the case of a manifold.     

The asymptotic behaviour of certain additive functionals of Brownian motion

We prove pathwise asymptotic stability for certain additive functionals of one- and two-dimensional Brownian motion.     

Almost sure convergence of the Bartlett estimator

Summary We study the almost sure convergence of the Bartlett estimator for the asymptotic variance of the sample mean of a stationary weekly dependent process. We also study the a.\ s.\ behavior of this estimator in the case of long-range dependent observations. In the weakly dependent case, we establish conditions under which the estimator is strongly consistent. We also show that, after appropriate normalization, the estimator converges a.s. in the long-range dependent case as well. In both cases, our conditions involve fourth order cumulants and assumptions on the rate of growth of the truncation parameter appearing in the definition of the Bartlett estimator.     

Almost sure convergence of branching processes

A general theorem concerning the almost sure convergence of some nonhomogeneous Markov chains, whose conditional distributions satisfy a certain convergence condition, is given. This result applied to branching processes with infinite mean yields almost sure convergence for a large class of processes converging in distribution, as well as a characterization of the limiting distribution function.     

Almost sure convergence of sample range

Let X ₁, X ₂, ... be i.i.d. random variables. The sample range is R _n = max {X _i , 1 ≤ i ≤ n} − min {X _i , 1 ≤ i ≤ n}. If for a non-degenerate distribution G and some sequences (α _k ), (β _k ) then we have

Almost sure convergence of the kT-occupation time density estimator

In this Note, we introduce an extension of the k-nearest neighbor estimator in continuous time, the $k_T$-occupation time estimator, and we give sufficient conditions for its existence. Then, we show the almost sure convergence for α-mixing and bounded processes in two cases, the superoptimal case (when parametric rates are reached) and the optimal case (when i.i.d. rates of density estimation are reached). To cite this article: B. Labrador, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Almost sure convergence of weighted partial sums

Sufficient conditions of covariance type are presented for weighted averages of random variables with arbitrary dependence structure to converge to 0, both for logarithmic and general weighting. As an application, an a.s. local limit theorem of Csáki, Földes and Révész is revisited and slightly improved.     

Almost sure asymptotics for the continuous parabolic Anderson model

We consider the parabolic Anderson problem ∂ _t u = κΔu + ξ(x)u on ℝ₊×ℝ ^d with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order term of the almost sure asymptotics of u(t, 0) as t→∞. Received: 26 July 1999 / Revised version: 6 April 2000 / Published online: 22 November 2000     

Almost sure local limit theorem for the Dickman distribution

We study the asymptotic behavior, and more precisely the second order properties, of the probabilistic model introduced in Hwang and Tsai (Comb Probab Comput 11(4):353–371, 2002) for describing the Dickman distribution. This model appears as an extremal example in the theory of the local and almost sure local limit theorem. We establish a delicate correlation inequality for this system. We apply it to obtain a fine almost sure local limit theorem. In doing so, we also give a corrected proof of the corresponding local limit theorem stated in Hwang and Tsai (Comb Probab Comput 11(4):353–371, 2002).     

Almost sure convergence of continuous time branching processes

We prove necessary and sufficient conditions for the transience of the non-zero states in a non-homogeneous, continuous time Markov branching process. The result is obtained by passing from results about the discrete time skeleton of the continuous time chain to the continuous time chain itself. An alternative proof of a result for continuous time Markov branching processes in random environments is then given, showing that earlier moment conditions were not necessary.