Weighted multiple ergodic averages and correlation sequences

We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good universal weight for mean convergence of such averages if and only if the average of this sequence times any nilsequence converges. Two decomposition results of independent interest play key roles in the proof. The first states that every bounded sequence in several variables satisfying some regularity conditions is a sum of a nilsequence and a sequence that has small uniformity norm (this generalizes a result of the second author and Kra); and the second states that every multiple correlation sequence in several variables is a sum of a nilsequence and a sequence that is small in uniform density (this generalizes a result of the first author). Furthermore, we use these results in order to establish mean convergence and recurrence results for a variety of sequences of dynamical and arithmetic origin and give some combinatorial implications.

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Integer part polynomial correlation sequences

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.67 ◽

2016 ◽

Vol 38 (4) ◽

pp. 1525-1542 ◽

Cited By ~ 1

Author(s):

ANDREAS KOUTSOGIANNIS

Keyword(s):

Error Term ◽

Uniform Density ◽

Integer Part ◽

Ergodic Averages ◽

Multiple Correlation ◽

Transference Principle ◽

The Mean ◽

Correlation Sequences ◽

Measure Preserving Actions ◽

Invent Math

Following an approach presented by Frantzikinakis [Multiple correlation sequences and nilsequences. Invent. Math. 202(2) (2015), 875–892], we prove that any multiple correlation sequence defined by invertible measure preserving actions of commuting transformations with integer part polynomial iterates is the sum of a nilsequence and an error term, which is small in uniform density. As an intermediate result, we show that multiple ergodic averages with iterates given by the integer part of real-valued polynomials converge in the mean. Also, we show that under certain assumptions the limit is zero. A transference principle, communicated to us by M. Wierdl, plays an important role in our arguments by allowing us to deduce results for $\mathbb{Z}$-actions from results for flows.

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Multiple polynomial correlation sequences and nilsequences

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385709000303 ◽

2009 ◽

Vol 30 (3) ◽

pp. 841-854 ◽

Cited By ~ 12

Author(s):

A. LEIBMAN

Keyword(s):

Uniform Density ◽

Uniform Limit ◽

Polynomial Sequence ◽

Several Variables ◽

Correlation Sequences

AbstractA basic nilsequence is a sequence of the form ψ(n)=f(Tnx), where x is a point of a compact nilmanifold X, T is a translation on X, and f∈C(X); a nilsequence is a uniform limit of basic nilsequences. Let X=G/Γ be a compact nilmanifold, Y be a subnilmanifold of X, g(n) be a polynomial sequence in G, and f∈C(X); we show that the sequence ∫ g(n)Yf, n∈ℤ, is the sum of a basic nilsequence and a sequence that converges to zero in uniform density. This implies that, given an ergodic invertible measure-preserving system (W,ℬ,μ,T), with μ(W)<∞, polynomials p1,…,pk∈ℤ[n], and sets A1,…,Ak∈ℬ, the sequence μ(Tp1(n)A1∩⋯∩Tpk(n)Ak) is the sum of a nilsequence and a sequence that converges to zero in uniform density. We also obtain a version of this result for the case where pi are polynomials in several variables.

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Nilsequences, null-sequences, and multiple correlation sequences

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2013.36 ◽

2013 ◽

Vol 35 (1) ◽

pp. 176-191 ◽

Cited By ~ 10

Author(s):

A. LEIBMAN

Keyword(s):

Finite Measure ◽

Null Sequence ◽

Uniform Density ◽

Multiple Correlation ◽

Uniform Limit ◽

Polynomial Sequence ◽

Correlation Sequences

AbstractA ($d$-parameter) basic nilsequence is a sequence of the form $\psi (n)= f({a}^{n} x)$,$n\in { \mathbb{Z} }^{d} $, where $x$ is a point of a compact nilmanifold $X$, $a$ is a translation on $X$, and$f\in C(X)$; a nilsequence is a uniform limit of basic nilsequences. If $X= G/ \Gamma $ is a compact nilmanifold, $Y$ is a subnilmanifold of $X$, $\mathop{(g(n))}\nolimits_{n\in { \mathbb{Z} }^{d} } $ is a polynomial sequence in $G$, and $f\in C(X)$, we show that the sequence $\phi (n)= \int \nolimits \nolimits_{g(n)Y} f$ is the sum of a basic nilsequence and a sequence that converges to zero in uniform density (a null-sequence). We also show that an integral of a family of nilsequences is a nilsequence plus a null-sequence. We deduce that for any invertible finite measure preserving system $(W, \mathcal{B} , \mu , T)$, polynomials ${p}_{1} , \ldots , {p}_{k} : { \mathbb{Z} }^{d} \longrightarrow \mathbb{Z} $, and sets ${A}_{1} , \ldots , {A}_{k} \in \mathcal{B} $, the sequence $\phi (n)= \mu ({T}^{{p}_{1} (n)} {A}_{1} \cap \cdots \cap {T}^{{p}_{k} (n)} {A}_{k} )$, $n\in { \mathbb{Z} }^{d} $, is the sum of a nilsequence and a null-sequence.

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Limit theorems for point processes generated in a general branching process

Advances in Applied Probability ◽

10.1017/s0001867800036430 ◽

1981 ◽

Vol 13 (04) ◽

pp. 650-668 ◽

Cited By ~ 2

Author(s):

Martin Härnqvist

Keyword(s):

Poisson Process ◽

Point Processes ◽

Branching Processes ◽

Special Problem ◽

Convergence Theory ◽

Regularity Conditions ◽

Convergence Results ◽

Mixed Poisson Process ◽

Proper Scaling ◽

Extra Point

With the general convergence theory for branching processes as basis a special problem is studied. An extra point process of events during life is assigned to each realised individual, and the behaviour of the superposition of such point processes in action is studied as the population grows. With the proper scaling and under some regularity conditions the superposition is shown to converge in distribution to a Poisson process. Another scaling gives rise to a mixed Poisson process as limit. Established weak convergence techniques for point processes are applied, together with some recent strong convergence results for branching processes.

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Measure change in multitype branching

Advances in Applied Probability ◽

10.1239/aap/1086957585 ◽

2004 ◽

Vol 36 (2) ◽

pp. 544-581 ◽

Cited By ~ 37

Author(s):

J. D. Biggins ◽

A. E. Kyprianou

Keyword(s):

Branching Processes ◽

Sufficient Conditions ◽

Branching Random Walk ◽

Stochastic Domination ◽

Moment Conditions ◽

Branching Brownian Motion ◽

Mean Convergence ◽

Convergence Results ◽

Watson Process ◽

The One

The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton-Watson processes with a general type space. Many examples satisfy stochastic domination conditions on the offspring distributions and suitable domination conditions combine nicely with general conditions for mean convergence to produce moment conditions, like the X log X condition of the Kesten-Stigum theorem. A general treatment of this phenomenon is given. The application of the approach to various branching processes is indicated. However, the main reason for developing the theory is to obtain martingale convergence results in a branching random walk that do not seem readily accessible with other techniques. These results, which are natural extensions of known results for martingales associated with binary branching Brownian motion, form the main application.

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High-entropy dual functions over finite fields and locally decodable codes

Forum of Mathematics Sigma ◽

10.1017/fms.2021.1 ◽

2021 ◽

Vol 9 ◽

Author(s):

Jop Briët ◽

Farrokh Labib

Keyword(s):

Finite Field ◽

Finite Fields ◽

Multiple Correlation ◽

High Entropy ◽

Polynomial Phase ◽

Locally Decodable Codes ◽

Phase Functions ◽

Correlation Sequences ◽

Locally Decodable ◽

Dual Functions

Abstract We show that for infinitely many primes p there exist dual functions of order k over ${\mathbb{F}}_p^n$ that cannot be approximated in $L_\infty $ -distance by polynomial phase functions of degree $k-1$ . This answers in the negative a natural finite-field analogue of a problem of Frantzikinakis on $L_\infty $ -approximations of dual functions over ${\mathbb{N}}$ (a.k.a. multiple correlation sequences) by nilsequences.

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Estimating Mahler's measure

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700013976 ◽

1995 ◽

Vol 51 (1) ◽

pp. 145-151

Author(s):

G.R. Everest

Keyword(s):

Linear Forms In Logarithms ◽

Algebraic Numbers ◽

Linear Forms ◽

Riemann Sums ◽

Integer Polynomials ◽

Several Variables ◽

Logarithmic Integral ◽

Baker's Theorem

In 1962, Mahler defined a measure for integer polynomials in several variables as the logarithmic integral over the torus. Many results exist about the values taken by the measure but many unsolved problems remain. In one variable, it is possible to express the measure as an effective limit of Riemann sums. We show that the same is true in several variables, using a non-obvious parametrisation of the torus together with Baker's Theorem on linear forms in logarithms of algebraic numbers.

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ASYMPTOTIC THEORY FOR SPECTRAL DENSITY ESTIMATES OF GENERAL MULTIVARIATE TIME SERIES

Econometric Theory ◽

10.1017/s0266466617000068 ◽

2017 ◽

Vol 34 (1) ◽

pp. 1-22 ◽

Cited By ~ 8

Author(s):

Wei Biao Wu ◽

Paolo Zaffaroni

Keyword(s):

Time Series ◽

Spectral Density ◽

Multivariate Time Series ◽

Rates Of Convergence ◽

Stationary Time Series ◽

Regularity Conditions ◽

Dynamic Factor ◽

Asymptotic Results ◽

Density Estimates ◽

Convergence Results

We derive uniform convergence results of lag-window spectral density estimates for a general class of multivariate stationary processes represented by an arbitrary measurable function of iid innovations. Optimal rates of convergence, that hold as both the time series and the cross section dimensions diverge, are obtained under mild and easily verifiable conditions. Our theory complements earlier results, most of which are univariate, which primarily concern in-probability, weak or distributional convergence, yet under a much stronger set of regularity conditions, such as linearity in iid innovations. Based on cross spectral density functions, we then propose a new test for independence between two stationary time series. We also explain the extent to which our results provide the foundation to derive the double asymptotic results for estimation of generalized dynamic factor models.

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