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On the conditional intensity of a random measure

Pierre Jacob, Paulo Eduardo Oliveira (1994)

Commentationes Mathematicae Universitatis Carolinae

We prove the existence of the conditional intensity of a random measure that is absolutely continuous with respect to its mean; when there exists an L p -intensity, p > 1 , the conditional intensity is obtained at the same time almost surely and in the mean.

On the convergence of moments in the almost sure central limit theorem for stochastic approximation algorithms

Peggy Cénac (2013)

ESAIM: Probability and Statistics

We study the almost sure asymptotic behaviour of stochastic approximation algorithms for the search of zero of a real function. The quadratic strong law of large numbers is extended to the powers greater than one. In other words, the convergence of moments in the almost sure central limit theorem (ASCLT) is established. As a by-product of this convergence, one gets another proof of ASCLT for stochastic approximation algorithms. The convergence result is applied to several examples as estimation...

On the convergence of moments in the CLT for triangular arrays with an application to random polynomials

Christophe Cuny, Michel Weber (2006)

Colloquium Mathematicae

We give a proof of convergence of moments in the Central Limit Theorem (under the Lyapunov-Lindeberg condition) for triangular arrays, yielding a new estimate of the speed of convergence expressed in terms of νth moments. We also give an application to the convergence in the mean of the pth moments of certain random trigonometric polynomials built from triangular arrays of independent random variables, thereby extending some recent work of Borwein and Lockhart.

On the convergence of the ensemble Kalman filter

Jan Mandel, Loren Cobb, Jonathan D. Beezley (2011)

Applications of Mathematics

Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables, the continuous mapping theorem gives convergence in probability of the ensemble members, and L p bounds on the ensemble then give L p convergence.

On the coupling property of Lévy processes

René L. Schilling, Jian Wang (2011)

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

We give necessary and sufficient conditions guaranteeing that the coupling for Lévy processes (with non-degenerate jump part) is successful. Our method relies on explicit formulae for the transition semigroup of a compound Poisson process and earlier results by Mineka and Lindvall–Rogers on couplings of random walks. In particular, we obtain that a Lévy process admits a successful coupling, if it is a strong Feller process or if the Lévy (jump) measure has an absolutely continuous component.

On the distance between ⟨X⟩ and L in the space of continuous BMO-martingales

Litan Yan, Norihiko Kazamaki (2005)

Studia Mathematica

Let X = (Xₜ,ℱₜ) be a continuous BMO-martingale, that is, | | X | | B M O s u p T | | E [ | X - X T | | T ] | | < , where the supremum is taken over all stopping times T. Define the critical exponent b(X) by b ( X ) = b > 0 : s u p T | | E [ e x p ( b ² ( X - X T ) ) | T ] | | < , where the supremum is taken over all stopping times T. Consider the continuous martingale q(X) defined by q ( X ) = E [ X | ] - E [ X | ] . We use q(X) to characterize the distance between ⟨X⟩ and the class L of all bounded martingales in the space of continuous BMO-martingales, and we show that the inequalities 1 / 4 d ( q ( X ) , L ) b ( X ) 4 / d ( q ( X ) , L ) hold for every continuous BMO-martingale X.

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