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,L_1-and L_2-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration

Under a first order moment condition on the immigration mechanism, we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(CBI process) converges almost surely. If an x log(x) moment condition on the branching mechanism does not hold, then the limit is zero. If this x log(x) moment condition holds, then we prove L_1 convergence as well. The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing.If, in addition, a suitable extra power moment condition on the branching mechanism holds, then we provide the correct scaling for the projection of a CBI process on certain left non-Perron eigenvectors of the branching mean matrix in order to have almost sure and L_1 limit. Moreover, under a second order moment condition on the branching and immigration mechanisms, we prove L_2 convergence of an appropriately scaled process and the above-mentioned projections as well. A representation of the limits is also provided under the same moment conditions.     

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 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 and bounded entropy

It is shown that the almost sure convergence property for certain sequences of operators {S _n{ implies a uniform bound on the metrical entropy of the sets {S _nf|n=1, 2, ...{, wheref is taken in theL ²-unit ball. This criterion permits one to unify certain counterexamples due to W. Rudin [Ru] and J.M. Marstrand [Mar] and has further applications. The theory of Gaussian processes is crucial in our approach.     

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 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.     

On long time almost sure asymptotics of renormalized branching diffusion processes

We are concerned with the long time behavior of branching diffusion processes. We give a partial answer to the following question: given a smooth density g₀, a branching rate and a spatial motion, does there exist a (nonspatially homogeneous) binary offspring distribution such that the corresponding renormalized branching process tends a.s. to g₀(y) dy as time grows to infinity?     

Almost sure convergence theorems in von Neumann algebras

The subadditive sequences of operators which belong to a von Neumann algebra with a faithful normal state and a given positive linear kernel are considered. We prove the almost sure convergence in Egorov’s sense for such sequences. Partially supported by the Ministry of Science and Ministry of Absorption. Partially sponsored by a grant from the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Almost sure convergence of stochastic taylor expansions for functions of real-valued two-parameter continuous brownian semimartingales

We show the convergence (almost sure and in quadratic mean) of the Taylor expansion     

Limit theorems for multitype continuous time Markov branching processes

Summary Let X(t)=(X ₁ (t), X ₂ (t), , X _t (t)) be a k-type (2k<) continuous time, supercritical, nonsingular, positively regular Markov branching process. Let M(t)=((m _ij (t))) be the mean matrix where m _ij (t)=E(X _j (t)¦X _r (0)= _ir for r=1, 2, , k) and write M(t)=exp(At). Let be an eigenvector of A corresponding to an eigenvalue . Assuming second moments this paper studies the limit behavior as t of the stochastic process . It is shown that i) if 2 Re >₁, then · X(t)e{–t¦ converges a.s. and in mean square to a random variable. ii) if 2 Re ₁ then [ · X(t)] f(v · X(t)) converges in law to a normal distribution where f(x)=(x) ^–1 if 2 Re <₁ and f(x)=(x log x)^–1 if 2 Re =₁, ₁ the largest real eigenvalue of A and v the corresponding right eigenvector.Research supported in part under contracts N0014-67-A-0112-0015 and NIH USPHS 10452 at Stanford University.     

Almost sure convergence of smoothingD m -splines for noisy data

Summary The purpose of this paper is to study the convergence of smoothingD ^m -splines relative to sets of data perturbed by a random noise. Conditions of almost sure convergence and error estimates are given.