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An algebraic approach to Pólya processes

Nicolas Pouyanne (2008)

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

Pólya processes are natural generalizations of Pólya–Eggenberger urn models. This article presents a new approach of their asymptotic behaviour via moments, based on the spectral decomposition of a suitable finite difference transition operator on polynomial functions. Especially, it provides new results for large processes (a Pólya process is called small when 1 is a simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part ≤1/2; otherwise, it is called large).

Asymptotics for the L p -deviation of the variance estimator under diffusion

Paul Doukhan, José R. León (2004)

ESAIM: Probability and Statistics

We consider a diffusion process X t smoothed with (small) sampling parameter ε . As in Berzin, León and Ortega (2001), we consider a kernel estimate α ^ ε with window h ( ε ) of a function α of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the L p deviations such as 1 h h ε p 2 α ^ ε - α p p - 𝔼 α ^ ε - α p p .

Asymptotics for the Lp-deviation of the variance estimator under diffusion

Paul Doukhan, José R. León (2010)

ESAIM: Probability and Statistics

We consider a diffusion process Xt smoothed with (small) sampling parameter ε. As in Berzin, León and Ortega (2001), we consider a kernel estimate α ^ ε with window h(ε) of a function α of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the Lp deviations such as 1 h h ε p 2 α ^ ε - α p p - I E α ^ ε - α p p .

Complete convergence in mean for double arrays of random variables with values in Banach spaces

Ta Cong Son, Dang Hung Thang, Le Van Dung (2014)

Applications of Mathematics

The rate of moment convergence of sample sums was investigated by Chow (1988) (in case of real-valued random variables). In 2006, Rosalsky et al. introduced and investigated this concept for case random variable with Banach-valued (called complete convergence in mean of order p ). In this paper, we give some new results of complete convergence in mean of order p and its applications to strong laws of large numbers for double arrays of random variables taking values in Banach spaces.

Constructive quantization: approximation by empirical measures

Steffen Dereich, Michael Scheutzow, Reik Schottstedt (2013)

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

In this article, we study the approximation of a probability measure μ on d by its empirical measure μ ^ N interpreted as a random quantization. As error criterion we consider an averaged p th moment Wasserstein metric. In the case where 2 p l t ; d , we establish fine upper and lower bounds for the error, ahigh resolution formula. Moreover, we provide a universal estimate based on moments, a Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions....

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