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The purpose of this paper is to investigate moderate deviations for the Durbin–Watson statistic associated with the stable first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We first establish a moderate deviation principle 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. It enables us to provide a moderate deviation...

Moment Inequality for the Martingale Square Function

Adam Osękowski (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Consider the sequence ${(C ₙ)}_{n \geq 1}$ of positive numbers defined by C₁ = 1 and $C_{n + 1} = 1 + C ₙ ² / 4$ , n = 1,2,.... Let M be a real-valued martingale and let S(M) denote its square function. We establish the bound |Mₙ|≤ Cₙ Sₙ(M), n=1,2,..., and show that for each n, the constant Cₙ is the best possible.

Monotonic rearrangements of functions with small mean oscillation

Dmitriy M. Stolyarov, Vasily I. Vasyunin, Pavel B. Zatitskiy (2015)

Studia Mathematica

We obtain sharp bounds for the monotonic rearrangement operator from "dyadic-type" classes to "continuous" ones; in particular, for the BMO space and Muckenhoupt classes. The idea is to connect the problem with a simple geometric construction named α-extension.

Noncommutative fractional integrals

Narcisse Randrianantoanina, Lian Wu (2015)

Studia Mathematica

Let ℳ be a hyperfinite finite von Nemann algebra and ${(ℳ_{k})}_{k \geq 1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration ${(ℳ_{k})}_{k \geq 1}$ . For a finite noncommutative martingale $x = {(x_{k})}_{1 \leq k \leq n} \subseteq L ₁ (ℳ)$ adapted to ${(ℳ_{k})}_{k \geq 1}$ and 0 < α < 1, the fractional integral of x of order α is defined by setting $I^{α} x = \sum_{k = 1}^{n} ζ_{k}^{α} d x_{k}$ for an appropriate sequence ${(ζ_{k})}_{k \geq 1}$ of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor equipped...

Noncommutative weak Orlicz spaces and martingale inequalities

Turdebek N. Bekjan, Zeqian Chen, Peide Liu, Yong Jiao (2011)

Studia Mathematica

This paper is devoted to the study of noncommutative weak Orlicz spaces and martingale inequalities. The Marcinkiewicz interpolation theorem is extended to include noncommutative weak Orlicz spaces as interpolation classes. As an application, we prove the weak type Φ-moment Burkholder-Gundy inequality for noncommutative martingales through establishing a weak type Φ-moment noncommutative Khinchin inequality for Rademacher random variables.

Note on the central limit theorem for stationary processes

Petrus Johannes Holewijn, Isaac Meilijson (1983)

Séminaire de probabilités de Strasbourg

On a relation between norms of the maximal function and the square function of a martingale

Masato Kikuchi (2013)

Colloquium Mathematicae

Let Ω be a nonatomic probability space, let X be a Banach function space over Ω, and let ℳ be the collection of all martingales on Ω. For $f = {(f ₙ)}_{n \in ℤ ₊} \in ℳ$ , let Mf and Sf denote the maximal function and the square function of f, respectively. We give some necessary and sufficient conditions for X to have the property that if f, g ∈ ℳ and ${| | M g | |}_{X} {\leq | | M f | |}_{X}$ , then ${| | S g | |}_{X} {\leq C | | S f | |}_{X}$ , where C is a constant independent of f and g.

On almost sure convergence of vector valued pramarts and multivalued pramarts.

Choukairi-Dini, Ahmed (1996)

Journal of Convex Analysis

On certain probabilities equivalent to coin-tossing, d'après Schachermayer

Samia Beghdadi-Sakrani, Michel Émery (1999)

Séminaire de probabilités de Strasbourg

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