Almost sure invariance principles for partial sums of weakly dependent random variables

Philipp, Walter; Stout, William

doi:10.1090/memo/0161

Cited by 320 publications

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“…As a result we obtain an estimation similar to (6), but the constant Ku-c 0 is now replaced by KK V or K-K v c 0 . The set A being finite, there are finitely many expressions…”

Section: Assumptions and Main Resultsmentioning

confidence: 92%

“…. , can be treated as a functional of some process (£ n ) n6 N and to check that the assumptions of theorem 7.1 of [6] are satisfied in this case.…”

mentioning

confidence: 99%

“…In view of theorem 2 the process (F°f') i=ou can be treated as a functional of 'label-process' (£ n ) n =o,i,...-Now by using theorems 2, 3 and 4, we check as in the proof of theorem 4 in [3] that the assumptions of theorem 7.1 of [6] are satisfied.…”

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confidence: 99%

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Almost sure invariance principle for some maps of an interval

Ziemian

1985

Ergod. Th. Dynam. Sys.

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Abstract. We prove an almost sure invariance principle and a central limit theorem for the process (F°/") B > 0 , where / is a map of an interval with a non-positive Schwarzian derivative whose trajectories of critical points stay far from the critical points, and F is a measurable function with bounded p-variation ( p > 1).The almost sure invariance principle implies the Log-log laws, integral tests and a distributional type of invariance principle for the process (F°f) n3:0 .

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“…As a result we obtain an estimation similar to (6), but the constant Ku-c 0 is now replaced by KK V or K-K v c 0 . The set A being finite, there are finitely many expressions…”

Section: Assumptions and Main Resultsmentioning

confidence: 92%

“…. , can be treated as a functional of some process (£ n ) n6 N and to check that the assumptions of theorem 7.1 of [6] are satisfied in this case.…”

mentioning

confidence: 99%

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Almost sure invariance principle for some maps of an interval

Ziemian

1985

Ergod. Th. Dynam. Sys.

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“…For weakly dependent variables, this type of result is a consequence of a theorem due to Philipp-Stout [109]. It gives conditions which imply that the ASIP holds.…”

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

confidence: 79%

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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“…Actually, much more than (1.2) and (1.3) is true: Philipp and Stout [12] proved that under (1.1) the partial sum process S(t) = S(t, x) = ¿^,k<t cos nkx (t > 0) is nearly Wiener in the sense that without changing its distribution it can be redefined on a suitable probability space together with a Wiener process {W(t), t > 0} such that (1.4) S(t) = W(t/2) + 0(tx/2~p) a.s. as/^oo for some constant p > 0. The approximation (1.4) implies not only the central limit theorem (1.2) and the law of the iterated logarithm (1.3) but it extends a large class of limit theorems of independent r.v.…”

Section: Introductionmentioning

confidence: 99%

Critical Lil Behavior of the Trigonometric System

Berkes¹

1993

Transactions of the American Mathematical Society

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Abstract. It is a classical fact that for rapidly increasing (nk) the sequence (cosrt^x) behaves like a sequence of i.i.d. random variables. Actually, this almost i.i.d. behavior holds if (nk) grows faster than ecv/* ; below this speed we have strong dependence. While there is a large literature dealing with the almost i.i.d. case, practically nothing is known on what happens at the critical speed nk ~ ec^k (critical behavior) and what is the probabilistic nature of (cos/î^x) in the strongly dependent domain. In our paper we study the critical LIL behavior of (cosnkx) i.e., we investigate how classical fluctuational theorems like the law of the iterated logarithm and the Kolmogorov-Feller test turn to nonclassical laws in the immediate neighborhood of nk ~ ecv^ .

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Almost sure invariance principles for partial sums of weakly dependent random variables

Cited by 320 publications

References 0 publications

Almost sure invariance principle for some maps of an interval

Almost sure invariance principle for some maps of an interval

Dynamics in Several Complex Variables

Critical Lil Behavior of the Trigonometric System

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