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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 sequences of iterates of random-valued vector functions

Rafał Kapica (2007)

Annales Polonici Mathematici

Given a probability space (Ω,,P) and a subset X of a normed space we consider functions f:X × Ω → X and investigate the speed of convergence of the sequence (fⁿ(x,·)) of the iterates f ⁿ : X × Ω ℕ → X defined by f¹(x,ω ) = f(x,ω₁), f n + 1 ( x , ω ) = f ( f ⁿ ( x , ω ) , ω n + 1 ) .

On the law of large numbers for continuous-time martingales and applications to statistics.

Hung T. Nguyen, Tuan D. Pham (1982)

Stochastica

In order to develop a general criterion for proving strong consistency of estimators in Statistics of stochastic processes, we study an extension, to the continuous-time case, of the strong law of large numbers for discrete time square integrable martingales (e.g. Neveu, 1965, 1972). Applications to estimation in diffusion models are given.

On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Alexander R. Pruss (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Let Ω be a countable infinite product Ω ₁ ℕ of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have E * [ X ₁ ] ≤ l i m i n f S ₙ / n ≤ l i m s u p S ₙ / n ≤ E * [ X ₁ ] , where E * and E* are the lower and upper expectations. We ask...

On the negative dependence in Hilbert spaces with applications

Nguyen Thi Thanh Hien, Le Van Thanh, Vo Thi Hong Van (2019)

Applications of Mathematics

This paper introduces the notion of pairwise and coordinatewise negative dependence for random vectors in Hilbert spaces. Besides giving some classical inequalities, almost sure convergence and complete convergence theorems are established. Some limit theorems are extended to pairwise and coordinatewise negatively dependent random vectors taking values in Hilbert spaces. An illustrative example is also provided.

On the strong convergence for weighted sums of asymptotically almost negatively associated random variables

Haiwu Huang, Guangming Deng, QingXia Zhang, Yuanying Jiang (2014)

Kybernetika

Applying the moment inequality of asymptotically almost negatively associated (AANA, in short) random variables which was obtained by Yuan and An (2009), some strong convergence results for weighted sums of AANA random variables are obtained without assumptions of identical distribution, which generalize and improve the corresponding ones of Zhou et al. (2011), Sung (2011, 2012) to the case of AANA random variables, respectively.

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