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A sharp analysis on the asymptotic behavior of the Durbin–Watson statistic for the first-order autoregressive process

Bernard Bercu, Frédéric Proïa (2013)

ESAIM: Probability and Statistics

The purpose of this paper is to provide a sharp analysis on the asymptotic behavior of the Durbin–Watson statistic. We focus our attention on the first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We establish the almost sure convergence and the asymptotic normality for both the least squares estimator of the unknown parameter of the autoregressive process as well as for the serial correlation estimator associated with the driven noise....

A Weak-Type Inequality for Submartingales and Itô Processes

Adam Osękowski (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate Y W ( 2 ( α + 1 ) ² ) / ( 2 α + 1 ) X L . Here W is the weak- L space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

Almost sure limit theorems for dependent random variables

Michał Seweryn (2010)

Banach Center Publications

For a sequence of dependent random variables ( X k ) k we consider a large class of summability methods defined by R. Jajte in [jaj] as follows: For a pair of real-valued nonnegative functions g,h: ℝ⁺ → ℝ⁺ we define a sequence of “weighted averages” 1 / g ( n ) k = 1 n ( X k ) / h ( k ) , where g and h satisfy some mild conditions. We investigate the almost sure behavior of such transformations. We also take a close look at the connection between the method of summation (that is the pair of functions (g,h)) and the coefficients that measure...

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).

An alternative proof of the uniqueness of martingale-coboundary decomposition of strictly stationary processes

Takehiko Morita (2019)

Commentationes Mathematicae Universitatis Carolinae

P. Samek and D. Volný, in the paper ``Uniqueness of a martingale-coboundary decomposition of a stationary processes" (1992), showed the uniqueness of martingale-coboundary decomposition of strictly stationary processes. The original proof is given by reducing the problem to the ergodic case. In this note we give another proof without such reduction.

An elementary proof of the Dalang-Morton-Willinger theorem

Armen Edigarian, Agnieszka Rygiel (2007)

Applicationes Mathematicae

L. C. G. Rogers has given an elementary proof of the fundamental theorem of asset pricing in the case of finite discrete time, due originally to Dalang, Morton and Willinger. The purpose of this paper is to give an even simpler proof of this important theorem without using the existence of regular conditional distribution, in contrast to Rogers' proof.

An extension of an inequality due to Stein and Lepingle

Ferenc Weisz (1996)

Colloquium Mathematicae

Hardy spaces consisting of adapted function sequences and generated by the q-variation and by the conditional q-variation are considered. Their dual spaces are characterized and an inequality due to Stein and Lepingle is extended.

Atomic decomposition of predictable martingale Hardy space with variable exponents

Zhiwei Hao (2015)

Czechoslovak Mathematical Journal

This paper is mainly devoted to establishing an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. More precisely, let ( Ω , , ) be a probability space and p ( · ) : Ω ( 0 , ) be a -measurable function such that 0 < inf x Ω p ( x ) sup x Ω p ( x ) < . It is proved that a predictable martingale Hardy space 𝒫 p ( · ) has an atomic decomposition by some key observations and new techniques. As an application, we obtain the boundedness of fractional integrals on the predictable martingale Hardy space with...

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