A note on the almost sure central limit theorem

Lacey, Michael T.; Philipp, Walter

doi:10.1016/0167-7152(90)90056-d

Cited by 212 publications

(133 citation statements)

References 6 publications

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“…The ASCLT was simultaneously proved by Brosamler [3] and Schatte [16], and, in its present form, by Lacey and Phillip [11]. In contrast with the wide literature on the ASCLT for independent random variables, very few references are available on the ASCLT for martingales, apart from the recent work of Bercu and Fort [1], [2] and the important contributions of Chaâbane and Maâouia [4], Chaâbane [5], Chaâbane and Touati [6], and Lifshits [14], [15].…”

Section: Introductionmentioning

confidence: 99%

On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

2009

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We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure central limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.

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Section: Introductionmentioning

confidence: 99%

On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

2009

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“…The present form appearing in the above definition was stated by Lacey and Philipp [10] in 1990. And in 1999, Ibragimov and Lifshits [6] gave the above sufficient condition.…”

Section: Ibragimov-lifshits Criterionmentioning

confidence: 99%

Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals

Zheng

2017

Stochastic Processes and their Applications

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In this work, we study the normal approximation and almost sure central limit theorems for some functionals of an independent sequence of Rademacher random variables. In particular, we provide a new chain rule that improves the one derived by Nourdin, Peccati and Reinert(2010) and then we deduce the bound on Wasserstein distance for normal approximation using the (discrete) Malliavin-Stein approach. Besides, we are able to give the almost sure central limit theorem for a sequence of random variables inside a fixed Rademacher chaos using the Ibragimov-Lifshits criterion.

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“…It gives conditions which imply that the ASIP holds. Lacey-Philipp proved in [95] that the ASIP implies the almost sure central limit theorem. For holomorphic endomorphisms of P k , we have the following result due to Dupont which holds in particular for Hölder continuous observables [58].…”

Section: Finite Positive Number Which Vanishes If and Only If ϕ Is A mentioning

confidence: 99%

Dynamics in Several Complex Variables

Abate

2017

Lecture Notes in Mathematics

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The emphasis of this introductory course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic (p.s.h.) functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. Nervertheless, we choose to show their effectiveness and to describe the theory for two large families of maps: the endomorphisms of projective spaces and the polynomial-like mappings.The first chapter deals with holomorphic endomorphisms of the projective space P k . We establish the first properties and give several constructions for the Green currents T p and the equilibrium measure µ = T k . The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems. We show the existence of a proper algebraic set E , totally invariant, i.e. f −1 (E ) = f (E ) = E , such that when a ∈ E , the probability measures, equidistributed on the fibers f −n (a), converge towards the equilibrium measure µ, as n goes to infinity. A similar result holds for the restriction of f to invariant subvarieties. We survey the equidistribution problem when points are replaced by varieties of arbitrary dimension, and discuss the equidistribution of periodic points. We then establish ergodic properties of µ: Kmixing, exponential decay of correlations for various classes of observables, central limit theorem and large deviations theorem. We heavily use the compactness of the space DSH(P k ) of differences of quasi-p.s.h. functions. In particular, we show that the measure µ is moderate, i.e. µ, e α|ϕ| ≤ c, on bounded sets of ϕ in DSH(P k ), for suitable positive constants α, c. Finally, we study the entropy, the Lyapounov exponents and the dimension of µ.The second chapter develops the theory of polynomial-like maps, i.e. proper holomorphic maps f : U → V where U, V are open subsets of C k with V convex and U ⋐ V . We introduce the dynamical degrees for such maps and construct the equilibrium measure µ of maximal entropy. Then, under a natural assumption on the dynamical degrees, we prove equidistribution properties of points and various statistical properties of the measure µ. The assumption is stable under small pertubations on the map. We also study the dimension of µ, the Lyapounov exponents and their variation.Our aim is to get a self-contained text that requires only a minimal background. In order to help the reader, an appendix gives the basics on p.s.h. functions, positive closed currents and super-potentials on projective spaces. Some exercises are proposed and an extensive bibliography is given.

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A note on the almost sure central limit theorem

Cited by 212 publications

References 6 publications

On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals

Dynamics in Several Complex Variables

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