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Emmanuel Lesigne

Publications and source records attributed to Emmanuel Lesigne.

14 recordsLinked to original sources
arXiv2024

What can be the limit in the CLT for a field of martingale differences?

The now classical convergence in distribution theorem for well normalized sums ofstationary martingale increments has been extended to multi-indexed martingaleincrements (see Volný (2019) and references in there). In the presentarticle we make progress in the identification of the limit law.In dimension one, as soon as the stationary martingale increments form an ergodic process, the limit law is normal, and it is stillthe case for multi-indexed martingale increments when one of the processes defined by one coordinate of the{\it multidimensional time} is ergodic. In the general case, the limit may be non normal.The dynamical properties of the $\mathbb{Z}^d$-measure preserving action associatedto the stationary random field allows us to give a necessary and sufficient conditionfor the existence of a non-normal limit law, in terms of entropy of some random processes.The identification of a {\it natural} factor on which the $\mathbb{Z}^d$-action is {\it of product type

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arXiv2015

Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths

We introduce and characterize central probability distributions on Littelmann paths. Next we establish a law of large numbers and a central limit theorem for the generalized Pitmann transform. We then study harmonic functions on multiplicative graphs defined from the tensor powers of finite-dimensional Lie algebras representations. Finally, we show there exists an inverse of the generalized Pitman transform defined almost surely on the set of infinite paths remaining in the Weyl chamber and explain how it can be computed.

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arXiv2013

Conditioned random walks from Kac-Moody root systems

Random paths are time continuous interpolations of random walks. By using Littelmann path model, we associate to each irreducible highest weight module of a Kac Moody algebra g a random path W. Under suitable hypotheses, we make explicit the probability of the event E: W never exits the Weyl chamber of g. We then give the law of the random walk defined by W conditioned by the event E and proves this law can be recovered by applying to W the generalized Pitmann transform introduced by Biane, Bougerol and O'Connell. This generalizes the main results of [10] and [16] to Kac Moody root systems and arbitrary highest weight modules. Moreover, we use here a completely new approach by exploiting the symmetry of our construction under the action of the Weyl group of g rather than renewal theory and Doob's theorem on Martin kernels.

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arXiv2013

A Random Field Model and its Application in Industrial Production

In competitive industries, a reliable yield forecasting is a prime factor to accurately determine the production costs and therefore ensure profitability. Indeed, quantifying the risks long before the effective manufacturing process enables fact-based decision-making. From the development stage, improvement efforts can be early identified and prioritized. In order to measure the impact of industrial process fluctuations on the product performances, the construction of a failure risk probability estimator is presented in this article. The complex relationship between the process technology and the product design (non linearities, multi-modal features...) is handled via random process regression. A random field encodes, for each product configuration, the available information regarding the risk of non-compliance. After a brief presentation of the Gaussian model approach, we describe a Bayesian reasoning avoiding a priori choices of location and scale parameters. The Gaussian mixture prior, conditioned by measured (or calculated) data, yields a posterior characterized by a multivariate Student distribution. The probabilistic nature of the model is then operated to derive a failure risk probability, defined as a random variable. To do this, our approach is to consider as random all unknown, inaccessible or fluctuating data. In order to propagate uncertainties, a fuzzy set approach provides an appropriate framework for the implementation of a Bayesian model mimicking expert elicitation. The underlying leitmotiv is to insert minimal a priori information in the failure risk model. The relevancy of this concept is illustrated with theoretical examples.

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arXiv2012

Conditioned one-way simple random walk and representation theory

We call one-way simple random walk a random walk in the quadrant Z_+^n whose increments belong to the canonical base. In relation with representation theory of Lie algebras and superalgebras, we describe the law of such a random walk conditioned to stay in a closed octant, a semi-open octant or other types of semi-groups. The combinatorial representation theory of these algebras allows us to describe a generalized Pitman transformation which realizes the conditioning on the set of paths of the walk. We pursue here in a direction initiated by O'Connell and his coauthors [13,14,2], and also developed in [12]. Our work relies on crystal bases theory and insertion schemes on tableaux described by Kashiwara and his coauthors in [1] and, very recently, in [5].

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arXiv2011

Random Sequences and Pointwise Convergence of Multiple Ergodic Averages

We prove pointwise convergence, as $N\to \infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f(T^nx)\cdot g(S^{a_n}x)$, where $T$ and $S$ are commuting measure preserving transformations, and $a_n$ is a random version of the sequence $[n^c]$ for some appropriate $c>1$. We also prove similar mean convergence results for averages of the form $\frac{1}{N}\sum_{n=1}^N f(T^{a_n}x)\cdot g(S^{a_n}x)$, as well as pointwise results when $T$ and $S$ are powers of the same transformation. The deterministic versions of these results, where one replaces $a_n$ with $[n^c]$, remain open, and we hope that our method will indicate a fruitful way to approach these problems as well.

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arXiv2010

Random walks in Weyl chambers and crystals

We use Kashiwara crystal basis theory to associate a random walk W to each irreducible representation V of a simple Lie algebra. This is achieved by endowing the crystal attached to V with a (possibly non uniform) probability distribution compatible with its weight graduation. We then prove that the generalized Pitmann transform defined by Biane, Bougerol and O'Connell for similar random walks with uniform distributions yields yet a Markov chain. When the representation is minuscule, and the associated random walk has a drift in the Weyl chamber, we establish that this Markov chain has the same law as W conditionned to never exit the cone of dominant weights. At the heart of our proof is a quotient version of a renewal theorem that we state in the context of general random walks in a lattice.

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arXiv2009

Powers of sequences and convergence of ergodic averages

A sequence $(s_n)$ of integers is good for the mean ergodic theorem if for each invertible measure preserving system $(X,\mathcal{B},μ,T)$ and any bounded measurable function $f$, the averages $ \frac1N \sum_{n=1}^N f(T^{s_n}x)$ converge in the $L^2$ norm. We construct a sequence $(s_n)$ that is good for the mean ergodic theorem, but the sequence $(s_n^2)$ is not. Furthermore, we show that for any set of bad exponents $B$, there is a sequence $(s_n)$ where $(s_n^k)$ is good for the mean ergodic theorem exactly when $k$ is not in $B$. We then extend this result to multiple ergodic averages. We also prove a similar result for pointwise convergence of single ergodic averages.

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arXiv2009

Averages on annuli of Eulidean space

We study the range of validity of differentiation theorems and ergodic theorems for $\R^d$ actions, for averages on "thick spheres" of Euclidean space.

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arXiv2008

Powers of sequences and recurrence

We study recurrence, and multiple recurrence, properties along the $k$-th powers of a given set of integers. We show that the property of recurrence for some given values of $k$ does not give any constraint on the recurrence for the other powers. This is motivated by similar results in number theory concerning additive basis of natural numbers. Moreover, motivated by a result of Kamae and Mendès-France, that links single recurrence with uniform distribution properties of sequences, we look for an analogous result dealing with higher order recurrence and make a related conjecture.

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arXiv2007

Van der Corput sets in Z^d

In this partly expository paper we study van der Corput sets in $\Z^d$, with a focus on connections with harmonic analysis and recurrence properties of measure preserving dynamical systems. We prove multidimensional versions of some classical results obtained for $d=1$ in \cite{K-MF} and \cite{R}, establish new characterizations, introduce and discuss some modifications of van der Corput sets which correspond to various notions of recurrence, provide numerous examples and formulate some natural open questions.

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arXiv2007

Intersective polynomials and polynomial Szemeredi theorem

Let $P=\{p_{1},\ld,p_{r}\}\subset\Q[n_{1},\ld,n_{m}]$ be a family of polynomials such that $p_{i}(\Z^{m})\sle\Z$, $i=1,\ld,r$. We say that the family $P$ has {\it PSZ property} if for any set $E\sle\Z$ with $d^{*}(E)=\limsup_{N-M\ras\infty}\frac{|E\cap[M,N-1]|}{N-M}>0$ there exist infinitely many $n\in\Z^{m}$ such that $E$ contains a polynomial progression of the form \hbox{$\{a,a+p_{1}(n),\ld,a+p_{r}(n)\}$}. We prove that a polynomial family $P=\{p_{1},\ld,p_{r}\}$ has PSZ property if and only if the polynomials $p_{1},\ld,p_{r}$ are {\it jointly intersective}, meaning that for any $k\in\N$ there exists $n\in\Z^{m}$ such that the integers $p_{1}(n),\ld,p_{r}(n)$ are all divisible by $k$. To obtain this result we give a new ergodic proof of the polynomial Szemerédi theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If $p_{1},\ld,p_{r}\in\Q[n]$ are jointly intersective integral polynomials, then for any finite partition of $\Z$, $\Z=\bigcup_{i=1}^{k}E_{i}$, there exist $i\in\{1,\ld,k\}$ and $a,n\in E_{i}$ such that $\{a,a+p_{1}(n),\ld,a+p_{r}(n)\}\sln E_{i}$.

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