We establish scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. These concern the asymptotic behaviour of the graph distance from the origin and the spatial location of the random walk in question. The limiting processes are the analogues of those for higher-dimensional versions of the model, but additional logarithmic terms in the scaling factors are needed to see these. The proof applies recently developed machinery relating the scaling of resistance metric spaces and stochastic processes, with key inputs being natural scaling statements for the random walk’s invariant measure, the associated effective resistance metric, the graph distance, and the cut times for the underlying simple random walk.
{"title":"Scaling limit for random walk on the range of random walk in four dimensions","authors":"D. Croydon, D. Shiraishi","doi":"10.1214/22-aihp1243","DOIUrl":"https://doi.org/10.1214/22-aihp1243","url":null,"abstract":"We establish scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. These concern the asymptotic behaviour of the graph distance from the origin and the spatial location of the random walk in question. The limiting processes are the analogues of those for higher-dimensional versions of the model, but additional logarithmic terms in the scaling factors are needed to see these. The proof applies recently developed machinery relating the scaling of resistance metric spaces and stochastic processes, with key inputs being natural scaling statements for the random walk’s invariant measure, the associated effective resistance metric, the graph distance, and the cut times for the underlying simple random walk.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"8 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79791705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There has recently been much activity within the Kardar-Parisi-Zhang universality class spurred by the construction of the canonical limiting object, the parabolic Airy sheet $mathcal{S}:mathbb{R}^2tomathbb{R}$ [arXiv:1812.00309]. The parabolic Airy sheet provides a coupling of parabolic Airy$_2$ processes -- a universal limiting geodesic weight profile in planar last passage percolation models -- and a natural goal is to understand this coupling. Geodesic geometry suggests that the difference of two parabolic Airy$_2$ processes, i.e., a difference profile, encodes important structural information. This difference profile $mathcal{D}$, given by $mathbb{R}tomathbb{R}:xmapsto mathcal{S}(1,x)- mathcal{S}(-1,x)$, was first studied by Basu, Ganguly, and Hammond [arXiv:1904.01717], who showed that it is monotone and almost everywhere constant, with its points of non-constancy forming a set of Hausdorff dimension $1/2$. Noticing that this is also the Hausdorff dimension of the zero set of Brownian motion leads to the question: is there a connection between $mathcal{D}$ and Brownian local time? Establishing that there is indeed a connection, we prove two results. On a global scale, we show that $mathcal{D}$ can be written as a Brownian local time patchwork quilt, i.e., as a concatenation of random restrictions of functions which are each absolutely continuous to Brownian local time (of rate four) away from the origin. On a local scale, we explicitly obtain Brownian local time of rate four as a local limit of $mathcal{D}$ at a point of increase, picked by a number of methods, including at a typical point sampled according to the distribution function $mathcal{D}$. Our arguments rely on the representation of $mathcal{S}$ in terms of a last passage problem through the parabolic Airy line ensemble and an understanding of geodesic geometry at deterministic and random times.
{"title":"Local and global comparisons of the Airy difference profile to Brownian local time","authors":"S. Ganguly, Milind Hegde","doi":"10.1214/22-aihp1290","DOIUrl":"https://doi.org/10.1214/22-aihp1290","url":null,"abstract":"There has recently been much activity within the Kardar-Parisi-Zhang universality class spurred by the construction of the canonical limiting object, the parabolic Airy sheet $mathcal{S}:mathbb{R}^2tomathbb{R}$ [arXiv:1812.00309]. The parabolic Airy sheet provides a coupling of parabolic Airy$_2$ processes -- a universal limiting geodesic weight profile in planar last passage percolation models -- and a natural goal is to understand this coupling. Geodesic geometry suggests that the difference of two parabolic Airy$_2$ processes, i.e., a difference profile, encodes important structural information. This difference profile $mathcal{D}$, given by $mathbb{R}tomathbb{R}:xmapsto mathcal{S}(1,x)- mathcal{S}(-1,x)$, was first studied by Basu, Ganguly, and Hammond [arXiv:1904.01717], who showed that it is monotone and almost everywhere constant, with its points of non-constancy forming a set of Hausdorff dimension $1/2$. Noticing that this is also the Hausdorff dimension of the zero set of Brownian motion leads to the question: is there a connection between $mathcal{D}$ and Brownian local time? Establishing that there is indeed a connection, we prove two results. On a global scale, we show that $mathcal{D}$ can be written as a Brownian local time patchwork quilt, i.e., as a concatenation of random restrictions of functions which are each absolutely continuous to Brownian local time (of rate four) away from the origin. On a local scale, we explicitly obtain Brownian local time of rate four as a local limit of $mathcal{D}$ at a point of increase, picked by a number of methods, including at a typical point sampled according to the distribution function $mathcal{D}$. Our arguments rely on the representation of $mathcal{S}$ in terms of a last passage problem through the parabolic Airy line ensemble and an understanding of geodesic geometry at deterministic and random times.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"118 2 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84301867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study large uniform random maps with one face whose genus grows linearly with the number of edges, which are a model of discrete hyperbolic geometry. In previous works, several hyperbolic geometric features have been investigated. In the present work, we study the number of short cycles in a uniform unicellular map of high genus, and we show that it converges to a Poisson distribution. As a corollary, we obtain the law of the systole of uniform unicellular maps in high genus. We also obtain the asymptotic distribution of the vertex degrees in such a map.
{"title":"Short cycles in high genus unicellular maps","authors":"S. Janson, B. Louf","doi":"10.1214/21-aihp1218","DOIUrl":"https://doi.org/10.1214/21-aihp1218","url":null,"abstract":"We study large uniform random maps with one face whose genus grows linearly with the number of edges, which are a model of discrete hyperbolic geometry. In previous works, several hyperbolic geometric features have been investigated. In the present work, we study the number of short cycles in a uniform unicellular map of high genus, and we show that it converges to a Poisson distribution. As a corollary, we obtain the law of the systole of uniform unicellular maps in high genus. We also obtain the asymptotic distribution of the vertex degrees in such a map.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"6 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74257758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We prove the existence of semi-infinite geodesics for Brownian last-passage percolation (BLPP). Specifically, on a single event of probability one, there exist semi-infinite geodesics started from every space- time point and traveling in every asymptotic direction. Properties of these geodesics include uniqueness for a fixed initial point and direction, non-uniqueness for fixed direction but random initial points, and coalescence of all geodesics traveling in a common, fixed direction. Along the way, we prove that for fixed northeast and southwest directions, there almost surely exist no bi-infinite geodesics in the given directions. The semi-infinite geodesics are constructed from Busemann functions. Our starting point is a result of Alberts, Rassoul-Agha and Simper that established Busemann functions for fixed points and directions. Out of this, we construct the global process of Busemann functions simultaneously for all initial points and directions, and then the family of semi-infinite Busemann geodesics. The uncountable space of the semi-discrete setting requires extra consideration and leads to new phenomena, compared to discrete models.
{"title":"Busemann process and semi-infinite geodesics in Brownian last-passage percolation","authors":"T. Seppalainen, Evan L. Sorensen","doi":"10.1214/22-aihp1245","DOIUrl":"https://doi.org/10.1214/22-aihp1245","url":null,"abstract":". We prove the existence of semi-infinite geodesics for Brownian last-passage percolation (BLPP). Specifically, on a single event of probability one, there exist semi-infinite geodesics started from every space- time point and traveling in every asymptotic direction. Properties of these geodesics include uniqueness for a fixed initial point and direction, non-uniqueness for fixed direction but random initial points, and coalescence of all geodesics traveling in a common, fixed direction. Along the way, we prove that for fixed northeast and southwest directions, there almost surely exist no bi-infinite geodesics in the given directions. The semi-infinite geodesics are constructed from Busemann functions. Our starting point is a result of Alberts, Rassoul-Agha and Simper that established Busemann functions for fixed points and directions. Out of this, we construct the global process of Busemann functions simultaneously for all initial points and directions, and then the family of semi-infinite Busemann geodesics. The uncountable space of the semi-discrete setting requires extra consideration and leads to new phenomena, compared to discrete models.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"47 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80724942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider i.i.d. last-passage percolation on $mathbb{Z}^2$ with weights having distribution $F$ and time-constant $g_F$. We provide an explicit condition on the large deviation rate function for independent sums of $F$ that determines when some adjacent Busemann function increments are negatively correlated. As an example, we prove that $operatorname{Bernoulli}(p)$ weights for $p>p^*$, ($p^* approx 0.6504$) satisfy this condition. We prove this condition by establishing a direct relationship between the negative correlations of adjacent Busemann increments and the dominance of the time-constant $g_F$ by the function describing the time-constant of last-passage percolation with exponential or geometric weights.
{"title":"Negative correlation of adjacent Busemann increments","authors":"Ian Alevy, Arjun Krishnan","doi":"10.1214/21-aihp1236","DOIUrl":"https://doi.org/10.1214/21-aihp1236","url":null,"abstract":"We consider i.i.d. last-passage percolation on $mathbb{Z}^2$ with weights having distribution $F$ and time-constant $g_F$. We provide an explicit condition on the large deviation rate function for independent sums of $F$ that determines when some adjacent Busemann function increments are negatively correlated. As an example, we prove that $operatorname{Bernoulli}(p)$ weights for $p>p^*$, ($p^* approx 0.6504$) satisfy this condition. We prove this condition by establishing a direct relationship between the negative correlations of adjacent Busemann increments and the dominance of the time-constant $g_F$ by the function describing the time-constant of last-passage percolation with exponential or geometric weights.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"34 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80286156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider a uniformly sampled random $d$-regular graph on $n$ vertices. If $d$ is fixed and $n$ goes to $infty$ then we can relate typical (large probability) properties of such random graph to a family of invariant random processes (called"typical"processes) on the infinite $d$-regular tree $T_d$. This correspondence between ergodic theory on $T_d$ and random regular graphs is already proven to be fruitful in both directions. This paper continues the investigation of typical processes with a special emphasis on entropy. We study a natural notion of micro-state entropy for invariant processes on $T_d$. It serves as a quantitative refinement of the notion of typicality and is tightly connected to the asymptotic free energy in statistical physics. Using entropy inequalities, we provide new sufficient conditions for typicality for edge Markov processes. We also extend these notions and results to processes on unimodular Galton-Watson random trees.
{"title":"Typicality and entropy of processes on infinite trees","authors":"'Agnes Backhausz, C. Bordenave, B. Szegedy","doi":"10.1214/21-aihp1233","DOIUrl":"https://doi.org/10.1214/21-aihp1233","url":null,"abstract":"Consider a uniformly sampled random $d$-regular graph on $n$ vertices. If $d$ is fixed and $n$ goes to $infty$ then we can relate typical (large probability) properties of such random graph to a family of invariant random processes (called\"typical\"processes) on the infinite $d$-regular tree $T_d$. This correspondence between ergodic theory on $T_d$ and random regular graphs is already proven to be fruitful in both directions. This paper continues the investigation of typical processes with a special emphasis on entropy. We study a natural notion of micro-state entropy for invariant processes on $T_d$. It serves as a quantitative refinement of the notion of typicality and is tightly connected to the asymptotic free energy in statistical physics. Using entropy inequalities, we provide new sufficient conditions for typicality for edge Markov processes. We also extend these notions and results to processes on unimodular Galton-Watson random trees.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"2012 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88165627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Questions regarding the continuity in κ of the SLE κ traces and maps appear very naturally in the study of SLE. In order to study the first question, we consider a natural coupling of SLE traces: for different values of κ we use the same Brownian motion. It is very natural to assume that with probability one, SLE κ depends continuously on κ . It is rather easy to show that SLE is continuous in the Carath´eodory sense, but showing that SLE traces are continuous in the uniform sense is much harder. In this note we show that for a given sequence κ j → κ ∈ (0 , 8 / 3), for almost every Brownian motion SLE κ traces converge locally uniformly. This result was also recently obtained by Friz, Tran and Yuan using different methods. In our analysis, we provide a constructive way to study the SLE κ traces for varying parameter κ ∈ (0 , 8 / 3). The argument is based on a new dynamical view on the approximation of SLE curves by curves driven by a piecewise square root approximation of the Brownian motion. The second question can be answered naturally in the framework of Rough Path Theory. Using this theory, we prove that the solutions of the backward Loewner Differential Equation driven by √ κB t when started away from the origin are continuous in the p -variation topology in the parameter κ , for all κ ∈ R + .
{"title":"Continuity in κ in SLEκ theory using a constructive method and Rough Path Theory","authors":"D. Beliaev, Terry Lyons, Vlad Margarint","doi":"10.1214/20-AIHP1084","DOIUrl":"https://doi.org/10.1214/20-AIHP1084","url":null,"abstract":"Questions regarding the continuity in κ of the SLE κ traces and maps appear very naturally in the study of SLE. In order to study the first question, we consider a natural coupling of SLE traces: for different values of κ we use the same Brownian motion. It is very natural to assume that with probability one, SLE κ depends continuously on κ . It is rather easy to show that SLE is continuous in the Carath´eodory sense, but showing that SLE traces are continuous in the uniform sense is much harder. In this note we show that for a given sequence κ j → κ ∈ (0 , 8 / 3), for almost every Brownian motion SLE κ traces converge locally uniformly. This result was also recently obtained by Friz, Tran and Yuan using different methods. In our analysis, we provide a constructive way to study the SLE κ traces for varying parameter κ ∈ (0 , 8 / 3). The argument is based on a new dynamical view on the approximation of SLE curves by curves driven by a piecewise square root approximation of the Brownian motion. The second question can be answered naturally in the framework of Rough Path Theory. Using this theory, we prove that the solutions of the backward Loewner Differential Equation driven by √ κB t when started away from the origin are continuous in the p -variation topology in the parameter κ , for all κ ∈ R + .","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"31 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76064898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Erratum: Central limit theorems for eigenvalues in a spiked population model [Annales de l’Institut Henri Poincaré – Probabilités et Statistiques 2008, Vol. 44, No. 3, 447–474]","authors":"Z. Bai, Jianfeng Yao","doi":"10.1214/20-aihp1078","DOIUrl":"https://doi.org/10.1214/20-aihp1078","url":null,"abstract":"","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"12 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86360737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Given a solution Y to a rough differential equation (RDE), a recent result ( Ann. Probab. 47 (2019) 1–60) extends the classical Itô-Stratonovich formula and provides a closed-form expression for (cid:2) Y ◦ d X − (cid:2) Y d X , i.e. the difference between the rough and Skorohod integrals of Y with respect to X , where X is a Gaussian process with finite p -variation less than 3. In this paper, we extend this result to Gaussian processes with finite p -variation such that 3 ≤ p < 4. The constraint this time is that we restrict ourselves to Volterra Gaussian processes with kernels satisfying a natural condition, which however still allows the result to encompass many standard examples, including fractional Brownian motion with Hurst parameter H > 14 . As an application we recover Itô formulas in the case where the vector fields of the RDE governing Y are commutative. Résumé. (2019) l’intégrale rugueuse et l’intégrale de Skorohod de Y par rapport à X , où X est un processus Gaussien avec p -variation plus petite que 3. Dans cet article, nous étendons ce résultat au cas de processus Gaussiens avec p -variation telle que 3 ≤ p < 4. La contrainte ici est que nous nous restreignons au cas de processus Gaussiens de type Volterra avec des noyaux satisfaisant une condition naturelle, ce qui permet néanmoins de traiter beaucoup d’exemples classiques incluant le cas du mouvement Brownien fractionnaire avec paramètre de Hurst H > 14 . Comme application, nous retrouvons la formule d’Itô dans le cas où les champs de vecteurs de la RDE gouvernant Y sont commutatifs.
． 给定一个粗糙微分方程(RDE)的解Y，最近的一个结果(Ann。Probab. 47(2019) 1-60)扩展了经典Itô-Stratonovich公式，并提供了(cid:2) Y◦d X−(cid:2) Y d X的封闭形式表达式，即Y对X的粗糙积分和Skorohod积分之差，其中X是一个有限p变差小于3的高斯过程。在本文中，我们将这一结果推广到具有有限p变分的高斯过程，使得3≤p < 4。这一次的约束是，我们将自己限制在具有满足自然条件的核的Volterra高斯过程中，然而，这仍然允许结果包含许多标准示例，包括Hurst参数H > 14的分数布朗运动。作为一个应用，我们恢复Itô公式的情况下，RDE控制Y的向量场是可交换的。的简历。(2019) l ' intacriale rugueuse et l ' intacriale de Skorohod de Y par rapport X, où X est un procsus Gaussien avec p -variation + petite que 3。在第2篇文章中，在p -变异区间3≤p < 4的情况下，nous samsamons和samsamons都有可能发生变化。这个contrainte ici que nous restreignons au cas de process(高斯过程)，高斯过程(高斯过程)，高斯过程(高斯过程)，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程，高斯过程。在Comme应用程序中，nous retrouvons的公式为'Itô dans的公式为où， les champs的向量为RDE治理Y的交换。
{"title":"Skorohod and rough integration for stochastic differential equations driven by Volterra processes","authors":"T. Cass, Nengli Lim","doi":"10.1214/20-AIHP1074","DOIUrl":"https://doi.org/10.1214/20-AIHP1074","url":null,"abstract":". Given a solution Y to a rough differential equation (RDE), a recent result ( Ann. Probab. 47 (2019) 1–60) extends the classical Itô-Stratonovich formula and provides a closed-form expression for (cid:2) Y ◦ d X − (cid:2) Y d X , i.e. the difference between the rough and Skorohod integrals of Y with respect to X , where X is a Gaussian process with finite p -variation less than 3. In this paper, we extend this result to Gaussian processes with finite p -variation such that 3 ≤ p < 4. The constraint this time is that we restrict ourselves to Volterra Gaussian processes with kernels satisfying a natural condition, which however still allows the result to encompass many standard examples, including fractional Brownian motion with Hurst parameter H > 14 . As an application we recover Itô formulas in the case where the vector fields of the RDE governing Y are commutative. Résumé. (2019) l’intégrale rugueuse et l’intégrale de Skorohod de Y par rapport à X , où X est un processus Gaussien avec p -variation plus petite que 3. Dans cet article, nous étendons ce résultat au cas de processus Gaussiens avec p -variation telle que 3 ≤ p < 4. La contrainte ici est que nous nous restreignons au cas de processus Gaussiens de type Volterra avec des noyaux satisfaisant une condition naturelle, ce qui permet néanmoins de traiter beaucoup d’exemples classiques incluant le cas du mouvement Brownien fractionnaire avec paramètre de Hurst H > 14 . Comme application, nous retrouvons la formule d’Itô dans le cas où les champs de vecteurs de la RDE gouvernant Y sont commutatifs.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"8 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83265718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we give rates of convergence, for minimal distances and for the uniform distance, between the law of partial sums of martingale differences and thelimiting Gaussian distribution. More precisely, denoting by $P_{X}$ the law of a random variable $X$ and by $G_{a}$ the normal distribution ${mathcal N} (0,a)$, we are interested by giving quantitative estimates for the convergence of $P_{S_n/sqrt{V_n}}$ to $G_1$, where $S_n$ is the partial sum associated with either martingale differences sequences or more general dependent sequences, and $V_n= {rm Var}(S_n)$. Applications to linear statistics, non stationary $rho$-mixing sequences and sequential dynamical systems are given.
{"title":"Rates of convergence in the central limit theorem for martingales in the non stationary setting","authors":"J. Dedecker, F. Merlevède, E. Rio","doi":"10.1214/21-aihp1182","DOIUrl":"https://doi.org/10.1214/21-aihp1182","url":null,"abstract":"In this paper, we give rates of convergence, for minimal distances and for the uniform distance, between the law of partial sums of martingale differences and thelimiting Gaussian distribution. More precisely, denoting by $P_{X}$ the law of a random variable $X$ and by $G_{a}$ the normal distribution ${mathcal N} (0,a)$, we are interested by giving quantitative estimates for the convergence of $P_{S_n/sqrt{V_n}}$ to $G_1$, where $S_n$ is the partial sum associated with either martingale differences sequences or more general dependent sequences, and $V_n= {rm Var}(S_n)$. Applications to linear statistics, non stationary $rho$-mixing sequences and sequential dynamical systems are given.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"44 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84238515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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