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On Riesz product measures ; mutual absolute continuity and singularity

Shelby J. Kilmer, Sadahiro Saeki (1988)

Annales de l'institut Fourier

We give some criteria for mutual absolute continuity and for singularity of Riesz product measures on locally compact abelian groups. The first section gives the definition of such a measure which is more general than the usual definition. The second section provides three sufficient conditions for one Riesz product measure to be absolutely continuous with respect to another. One of our results contains a theorem of Brown-Moran-Ritter as a special case. The final section deals with random Riesz...

On small deviations of Gaussian processes using majorizing measures

Michel J. G. Weber (2012)

Colloquium Mathematicae

We give two examples of periodic Gaussian processes, having entropy numbers of exactly the same order but radically different small deviations. Our construction is based on Knopp's classical result yielding existence of continuous nowhere differentiable functions, and more precisely on Loud's functions. We also obtain a general lower bound for small deviations using the majorizing measure method. We show by examples that our bound is sharp. We also apply it to Gaussian independent sequences and...

On strong laws for generalized L-statistics with dependent data

David Gilat, Roelof Helmers (1997)

Commentationes Mathematicae Universitatis Carolinae

It is pointed out that a strong law of large numbers for L-statistics established by van Zwet (1980) for i.i.d. sequences, remains valid for stationary ergodic data. When the underlying process is weakly Bernoulli, the result extends even to generalized L-statistics considered in Helmers et al. (1988).

On the asymptotic form of convex hulls of Gaussian random fields

Youri Davydov, Vygantas Paulauskas (2014)

Open Mathematics

We consider a centered Gaussian random field X = X t : t ∈ T with values in a Banach space 𝔹 defined on a parametric set T equal to ℝm or ℤm. It is supposed that the distribution of X t is independent of t. We consider the asymptotic behavior of closed convex hulls W n = convX t : t ∈ T n, where (T n) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b n)n≥1 with probability 1, (in the sense...

On the Borel-Cantelli Lemma and moments

S. Amghibech (2006)

Commentationes Mathematicae Universitatis Carolinae

We present some extensions of the Borel-Cantelli Lemma in terms of moments. Our result can be viewed as a new improvement to the Borel-Cantelli Lemma. Our proofs are based on the expansion of moments of some partial sums by using Stirling numbers. We also give a comment concerning the results of Petrov V.V., A generalization of the Borel-Cantelli Lemma, Statist. Probab. Lett. 67 (2004), no. 3, 233–239.

On the Brunk-Chung type strong law of large numbers for sequences of blockwise m-dependent random variables

Le Van Thanh (2006)

ESAIM: Probability and Statistics

For a sequence of blockwise m-dependent random variables {Xn,n ≥ 1}, conditions are provided under which lim n → ∞ ( ∑ i = 1 n X i ) / b n = 0 almost surely where {bn,n ≥ 1} is a sequence of positive constants. The results are new even when bn ≡ nr,r > 0. As special case, the Brunk-Chung strong law of large numbers is obtained for sequences of independent random variables. The current work also extends results of Móricz [Proc. Amer. Math. Soc.101 (1987) 709–715], and Gaposhkin [Teor. Veroyatnost. i Primenen. 39 (1994) 804–812]....

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...

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