Let η = (η1, …, ηN) be a multinomial random vector with parameters n = η1 + ⋯ + ηN and pm > 0, m = 1, …, N, p1 + ⋯ + pN = 1. We assume that N →∞ and maxpm → 0 as n →∞. The probabilities of large deviations for statistics of the form h1(η1) + ⋯ + hN(ηN) are studied, where hm(x) is a real-valued function of a non-negative integer-valued argument. The new large deviation results for the power-divergence statistics and its most popular special variants, as well as for several count statistics are derived as consequences of the general theorems.
{"title":"The probabilities of large deviations for a certain class of statistics associated with multinomial distribution","authors":"S. Mirakhmedov","doi":"10.1051/ps/2020020","DOIUrl":"https://doi.org/10.1051/ps/2020020","url":null,"abstract":"Let η = (η1, …, ηN) be a multinomial random vector with parameters n = η1 + ⋯ + ηN and pm > 0, m = 1, …, N, p1 + ⋯ + pN = 1. We assume that N →∞ and maxpm → 0 as n →∞. The probabilities of large deviations for statistics of the form h1(η1) + ⋯ + hN(ηN) are studied, where hm(x) is a real-valued function of a non-negative integer-valued argument. The new large deviation results for the power-divergence statistics and its most popular special variants, as well as for several count statistics are derived as consequences of the general theorems.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"235 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85651205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let ξn be the polygonal line partial sums process built on i.i.d. centered random variables Xi, i ≥ 1. The Bernstein-Kantorovich theorem states the equivalence between the finiteness of E|X1|max(2,r) and the joint weak convergence in C[0, 1] of n−1∕2ξn to a Brownian motion W with the moments convergence of E∥n−1/2ξn∥∞r to E∥W∥∞r. For 0 < α < 1∕2 and p (α) = (1 ∕ 2 - α) -1, we prove that the joint convergence in the separable Hölder space Hαo of n−1∕2ξn to W jointly with the one of E∥n−1∕2ξn∥αr to E∥W∥αr holds if and only if P(|X1| > t) = o(t−p(α)) when r < p(α) or E|X1|r < ∞ when r ≥ p(α). As an application we show that for every α < 1∕2, all the α-Hölderian moments of the polygonal uniform quantile process converge to the corresponding ones of a Brownian bridge. We also obtain the asymptotic behavior of the rth moments of some α-Hölderian weighted scan statistics where the natural border for α is 1∕2 − 1∕p when E|X1|p < ∞. In the case where the Xi’s are p regularly varying, we can complete these results for α > 1∕2 − 1∕p with an appropriate normalization.
{"title":"On Bernstein–Kantorovich invariance principle in Hölder spaces and weighted scan statistics","authors":"A. Račkauskas, Charles Suquet","doi":"10.1051/ps/2019027","DOIUrl":"https://doi.org/10.1051/ps/2019027","url":null,"abstract":"Let ξn be the polygonal line partial sums process built on i.i.d. centered random variables Xi, i ≥ 1. The Bernstein-Kantorovich theorem states the equivalence between the finiteness of E|X1|max(2,r) and the joint weak convergence in C[0, 1] of n−1∕2ξn to a Brownian motion W with the moments convergence of E∥n−1/2ξn∥∞r to E∥W∥∞r. For 0 < α < 1∕2 and p (α) = (1 ∕ 2 - α) -1, we prove that the joint convergence in the separable Hölder space Hαo of n−1∕2ξn to W jointly with the one of E∥n−1∕2ξn∥αr to E∥W∥αr holds if and only if P(|X1| > t) = o(t−p(α)) when r < p(α) or E|X1|r < ∞ when r ≥ p(α). As an application we show that for every α < 1∕2, all the α-Hölderian moments of the polygonal uniform quantile process converge to the corresponding ones of a Brownian bridge. We also obtain the asymptotic behavior of the rth moments of some α-Hölderian weighted scan statistics where the natural border for α is 1∕2 − 1∕p when E|X1|p < ∞. In the case where the Xi’s are p regularly varying, we can complete these results for α > 1∕2 − 1∕p with an appropriate normalization.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"76 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90288193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The origin of power-law behavior (also known variously as Zipf’s law) has been a topic of debate in the scientific community for more than a century. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. In a highly cited article, Mark Newman [Contemp. Phys. 46 (2005) 323–351] reviewed some of the empirical evidence for the existence of power-law forms, however underscored that even though many distributions do not follow a power law, quite often many of the quantities that scientists measure are close to a Zipf law, and hence are of importance. In this paper we engage a variant of Zipf’s law with a general urn problem. A collector wishes to collect m complete sets of N distinct coupons. The draws from the population are considered to be independent and identically distributed with replacement, and the probability that a type-j coupon is drawn is denoted by p j , j = 1, 2, …, N . Let T m (N ) the number of trials needed for this problem. We present the asymptotics for the expectation (five terms plus an error), the second rising moment (six terms plus an error), and the variance of T m (N ) (leading term) as N →∞ , when p j = a j / ∑j =2 N +1 a j , where a j = (ln j )−p , p > 0. begin{equation*} p_{j}=frac{a_{j}}{sum_{j=2}^{N+1} a_{j}}, ,,,text{where},,, a_{j}=left(ln jright)^{-p}, ,,p>0.end{equation*} pj=aj ∑ j=2N+1aj,whereaj= lnj-p,p>0. Moreover, we prove that T m (N ) (appropriately normalized) converges in distribution to a Gumbel random variable. These “log-Zipf” classes of coupon probabilities are not covered by the existing literature and the present paper comes to fill this gap. In the spirit of a recent paper of ours [ESAIM: PS 20 (2016) 367–399] we enlarge the classes for which the Dixie cup problem is solved w.r.t. its moments, variance, distribution.
幂律行为(也称为齐夫定律)的起源在科学界已经争论了一个多世纪。幂律广泛出现在物理学、生物学、地球和行星科学、经济学和金融学、计算机科学、人口学和社会科学中。在一篇被大量引用的文章中，马克·纽曼[当代][物理学46(2005)323-351]回顾了幂律形式存在的一些经验证据，但强调了即使许多分布不遵循幂律，科学家测量的许多量通常接近齐夫定律，因此很重要。本文将齐夫定律的一个变体与一般瓮问题联系起来。一个收藏家希望收集m套完整的N种不同的优惠券。认为从总体中抽取的券是独立的，具有替换的同分布，抽取到j型券的概率记为p j, j = 1,2，…，N。设T m (N)为这个问题所需的试验次数。当p j = a j /∑j = 2n + 1a j，其中a j = (ln j)−p, p > 0时，我们给出了期望(五项加一个误差)、第二次上升矩(六项加一个误差)和T m (N)(首项)方差为N→∞的渐近性。begin{equation*} p_{j}=frac{a_{j}}{sum_{j=2}^{N+1} a_{j}}, ,,,text{where},,, a_{j}=left(ln jright)^{-p}, ,,p>0.end{equation*} pj=aj∑j=2N+1aj，其中aj= lnj-p,p>0。此外，我们证明了T m (N)(适当归一化)在分布上收敛于一个Gumbel随机变量。这些息票概率的“log-Zipf”类未被现有文献所涵盖，本文填补了这一空白。本着我们最近的一篇论文[ESAIM: PS 20(2016) 367-399]的精神，我们扩大了迪克西杯问题解决的类别，包括它的矩、方差和分布。
{"title":"The logarithmic Zipf law in a general urn problem","authors":"Aristides V. Doumas, V. Papanicolaou","doi":"10.1051/ps/2020011","DOIUrl":"https://doi.org/10.1051/ps/2020011","url":null,"abstract":"The origin of power-law behavior (also known variously as Zipf’s law) has been a topic of debate in the scientific community for more than a century. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. In a highly cited article, Mark Newman [Contemp. Phys. 46 (2005) 323–351] reviewed some of the empirical evidence for the existence of power-law forms, however underscored that even though many distributions do not follow a power law, quite often many of the quantities that scientists measure are close to a Zipf law, and hence are of importance. In this paper we engage a variant of Zipf’s law with a general urn problem. A collector wishes to collect m complete sets of N distinct coupons. The draws from the population are considered to be independent and identically distributed with replacement, and the probability that a type-j coupon is drawn is denoted by p j , j = 1, 2, …, N . Let T m (N ) the number of trials needed for this problem. We present the asymptotics for the expectation (five terms plus an error), the second rising moment (six terms plus an error), and the variance of T m (N ) (leading term) as N →∞ , when p j = a j / ∑j =2 N +1 a j , where a j = (ln j )−p , p > 0. begin{equation*} p_{j}=frac{a_{j}}{sum_{j=2}^{N+1} a_{j}}, ,,,text{where},,, a_{j}=left(ln jright)^{-p}, ,,p>0.end{equation*} pj=aj ∑ j=2N+1aj,whereaj= lnj-p,p>0. Moreover, we prove that T m (N ) (appropriately normalized) converges in distribution to a Gumbel random variable. These “log-Zipf” classes of coupon probabilities are not covered by the existing literature and the present paper comes to fill this gap. In the spirit of a recent paper of ours [ESAIM: PS 20 (2016) 367–399] we enlarge the classes for which the Dixie cup problem is solved w.r.t. its moments, variance, distribution.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"57 1","pages":"275-293"},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73591282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider the nonparametric regression model Y ni = g (t ni ) + e i , i = 1, 2, …, n , n ≥ 1, where e i , 1 ≤ i ≤ n , are asymptotically negatively associated (ANA, for short) random variables. Under some appropriate conditions, the Berry-Esseen bound of the wavelet estimator of g (⋅) is established. In addition, some numerical simulations are provided in this paper. The results obtained in this paper generalize some corresponding ones in the literature.
考虑非参数回归模型Y ni = g (t ni) + e i, i = 1,2，…，n, n≥1，其中e i, 1≤i≤n为渐近负相关(简称ANA)随机变量。在一定条件下，建立了g(⋅)的小波估计量的Berry-Esseen界。此外，本文还进行了数值模拟。本文所得结果推广了文献中相应的结果。
{"title":"The Berry-Esseen bound of a wavelet estimator in non-randomly designed nonparametric regression model based on ANA errors","authors":"Xu-fei Tang, Xuejun Wang, Yi Wu, Fei Zhang","doi":"10.1051/PS/2019017","DOIUrl":"https://doi.org/10.1051/PS/2019017","url":null,"abstract":"Consider the nonparametric regression model Y ni = g (t ni ) + e i , i = 1, 2, …, n , n ≥ 1, where e i , 1 ≤ i ≤ n , are asymptotically negatively associated (ANA, for short) random variables. Under some appropriate conditions, the Berry-Esseen bound of the wavelet estimator of g (⋅) is established. In addition, some numerical simulations are provided in this paper. The results obtained in this paper generalize some corresponding ones in the literature.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"56 1","pages":"21-38"},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72671001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work links the conditional probability structure of Lancaster probabilities to a construction of reversible continuous-time Markov processes. Such a task is achieved by using the spectral expansion of the corresponding transition probabilities in order to introduce a continuous time dependence in the orthogonal representation inherent to Lancaster probabilities. This relationship provides a novel methodology to build continuous-time Markov processes via Lancaster probabilities. Particular cases of well-known models are seen to fall within this approach. As a byproduct, it also unveils new identities associated to well known orthogonal polynomials.
{"title":"Continuous-time Markov processes, orthogonal polynomials and Lancaster probabilities","authors":"R. H. Mena, Freddy Palma","doi":"10.1051/ps/2020004","DOIUrl":"https://doi.org/10.1051/ps/2020004","url":null,"abstract":"This work links the conditional probability structure of Lancaster probabilities to a construction of reversible continuous-time Markov processes. Such a task is achieved by using the spectral expansion of the corresponding transition probabilities in order to introduce a continuous time dependence in the orthogonal representation inherent to Lancaster probabilities. This relationship provides a novel methodology to build continuous-time Markov processes via Lancaster probabilities. Particular cases of well-known models are seen to fall within this approach. As a byproduct, it also unveils new identities associated to well known orthogonal polynomials.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"43 1","pages":"100-112"},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78400395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, by a probabilistic approach we prove that there exists a unique viscosity solution to obstacle problems of quasilinear parabolic PDEs combined with Neumann boundary conditions and algebra equations. The existence and uniqueness for adapted solutions of fully coupled forward-backward stochastic differential equations with reflections play a crucial role. Compared with existing works, in our result the spatial variable of solutions of PDEs lives in a region without convexity constraints, the second order coefficient of PDEs depends on the gradient of the solution, and the required conditions for the coefficients are weaker.
{"title":"A probabilistic approach to quasilinear parabolic PDEs with obstacle and Neumann problems","authors":"Lishun Xiao, Shengjun Fan, D. Tian","doi":"10.1051/ps/2019023","DOIUrl":"https://doi.org/10.1051/ps/2019023","url":null,"abstract":"In this paper, by a probabilistic approach we prove that there exists a unique viscosity solution to obstacle problems of quasilinear parabolic PDEs combined with Neumann boundary conditions and algebra equations. The existence and uniqueness for adapted solutions of fully coupled forward-backward stochastic differential equations with reflections play a crucial role. Compared with existing works, in our result the spatial variable of solutions of PDEs lives in a region without convexity constraints, the second order coefficient of PDEs depends on the gradient of the solution, and the required conditions for the coefficients are weaker.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"14 1","pages":"207-226"},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84698315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The current article is devoted to the study of a mean-field system of particles. The question that we are interested in is the behaviour of the exit-time of the first particle (and the one of any particle) from a domain D on ℝd as the diffusion coefficient goes to 0. We establish a Kramers’ type law. In other words, we show that the exit-time is exponentially equivalent to [see formula in PDF], HN being the exit-cost. We also show that this exit-cost converges to some quantity H.
{"title":"Exit-time of mean-field particles system","authors":"J. Tugaut","doi":"10.1051/ps/2019028","DOIUrl":"https://doi.org/10.1051/ps/2019028","url":null,"abstract":"The current article is devoted to the study of a mean-field system of particles. The question that we are interested in is the behaviour of the exit-time of the first particle (and the one of any particle) from a domain D on ℝd as the diffusion coefficient goes to 0. We establish a Kramers’ type law. In other words, we show that the exit-time is exponentially equivalent to [see formula in PDF], HN being the exit-cost. We also show that this exit-cost converges to some quantity H.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"27 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81682928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present paper concerns the parametric estimation for the fractional Gaussian noise in a high-frequency observation scheme. The sequence of Le Cam’s one-step maximum likelihood estimators (OSMLE) is studied. This sequence is defined by an initial sequence of quadratic generalized variations-based estimators (QGV) and a single Fisher scoring step. The sequence of OSMLE is proved to be asymptotically efficient as the sequence of maximum likelihood estimators but is much less computationally demanding. It is also advantageous with respect to the QGV which is not variance efficient. Performances of the estimators on finite size observation samples are illustrated by means of Monte-Carlo simulations.
{"title":"One-step estimation for the fractional Gaussian noise at high-frequency","authors":"A. Brouste, M. Soltane, I. Votsi","doi":"10.1051/PS/2020022","DOIUrl":"https://doi.org/10.1051/PS/2020022","url":null,"abstract":"The present paper concerns the parametric estimation for the fractional Gaussian noise in a high-frequency observation scheme. The sequence of Le Cam’s one-step maximum likelihood estimators (OSMLE) is studied. This sequence is defined by an initial sequence of quadratic generalized variations-based estimators (QGV) and a single Fisher scoring step. The sequence of OSMLE is proved to be asymptotically efficient as the sequence of maximum likelihood estimators but is much less computationally demanding. It is also advantageous with respect to the QGV which is not variance efficient. Performances of the estimators on finite size observation samples are illustrated by means of Monte-Carlo simulations.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"96 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83375356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we approximate the invariant distributionνof an ergodic Jump Diffusion driven by the sum of a Brownian motion and a Compound Poisson process with sub-Gaussian jumps. We first construct an Euler discretization scheme with decreasing time steps. This scheme is similar to those introduced in Lamberton and PagèsBernoulli8(2002) 367-405. for a Brownian diffusion and extended in F. Panloup,Ann. Appl. Probab.18(2008) 379-426. to a diffusion with Lévy jumps. We obtain a non-asymptoticquasiGaussian (asymptotically Gaussian) concentration bound for the difference between the invariant distribution and the empirical distribution computed with the scheme of decreasing time step along appropriate test functionsfsuch thatf−ν(f) is a coboundary of the infinitesimal generator.
{"title":"Approximation of the invariant distribution for a class of ergodic jump diffusions","authors":"A. Gloter, Igor Honoré, D. Loukianova","doi":"10.1051/PS/2020023","DOIUrl":"https://doi.org/10.1051/PS/2020023","url":null,"abstract":"In this article, we approximate the invariant distributionνof an ergodic Jump Diffusion driven by the sum of a Brownian motion and a Compound Poisson process with sub-Gaussian jumps. We first construct an Euler discretization scheme with decreasing time steps. This scheme is similar to those introduced in Lamberton and PagèsBernoulli8(2002) 367-405. for a Brownian diffusion and extended in F. Panloup,Ann. Appl. Probab.18(2008) 379-426. to a diffusion with Lévy jumps. We obtain a non-asymptoticquasiGaussian (asymptotically Gaussian) concentration bound for the difference between the invariant distribution and the empirical distribution computed with the scheme of decreasing time step along appropriate test functionsfsuch thatf−ν(f) is a coboundary of the infinitesimal generator.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"8 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73335260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Random forests were introduced by Breiman in 2001. We study theoretical aspects of both original Breiman’s random forests and a simplified version, the centred random forests. Under the independent and identically distributed hypothesis, Scornet, Biau and Vert proved the consistency of Breiman’s random forest, while Biau studied the simplified version and obtained a rate of convergence in the sparse case. However, the i.i.d hypothesis is generally not satisfied for example when dealing with time series. We extend the previous results to the case where observations are weakly dependent, more precisely when the sequences are stationary β−mixing.
{"title":"Random forests for time-dependent processes","authors":"Benjamin Goehry","doi":"10.1051/PS/2020015","DOIUrl":"https://doi.org/10.1051/PS/2020015","url":null,"abstract":"Random forests were introduced by Breiman in 2001. We study theoretical aspects of both original Breiman’s random forests and a simplified version, the centred random forests. Under the independent and identically distributed hypothesis, Scornet, Biau and Vert proved the consistency of Breiman’s random forest, while Biau studied the simplified version and obtained a rate of convergence in the sparse case. However, the i.i.d hypothesis is generally not satisfied for example when dealing with time series. We extend the previous results to the case where observations are weakly dependent, more precisely when the sequences are stationary β−mixing.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"9 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89291747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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