The paper investigates existence and uniqueness for a stochastic differential equation (SDE) with distributional drift depending on the law density of the solution. Those equations are known as McKean SDEs. The McKean SDE is interpreted in the sense of a suitable singular martingale problem. A key tool used in the investigation is the study of the corresponding Fokker-Planck equation.
{"title":"McKean SDEs with singular coefficients","authors":"Elena Issoglio, F. Russo","doi":"10.1214/22-aihp1293","DOIUrl":"https://doi.org/10.1214/22-aihp1293","url":null,"abstract":"The paper investigates existence and uniqueness for a stochastic differential equation (SDE) with distributional drift depending on the law density of the solution. Those equations are known as McKean SDEs. The McKean SDE is interpreted in the sense of a suitable singular martingale problem. A key tool used in the investigation is the study of the corresponding Fokker-Planck equation.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"74 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85887934","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}
Fadhel Ayed, M. Battiston, F. Camerlenghi, S. Favaro
. Given n samples from a population of individuals belonging to different types with unknown proportions, how do we estimate the probability of discovering a new type at the ( n + 1)-th draw? This is a classical problem in statistics, commonly referred to as the missing mass estimation problem. Recent results have shown: i) the impossibility of estimating the missing mass without imposing further assumptions on type’s proportions; ii) the consistency of the Good-Turing estimator of the missing mass under the assumption that the tail of type’s proportions decays to zero as a regularly varying function with parameter α ∈ (0 , 1); ii) the rate of convergence n − α/ 2 for the Good-Turing estimator under the class of α ∈ (0 , 1) regularly varying P . In this paper we introduce an alternative, and remarkably shorter, proof of the impossibility of a distribution-free estimation of the missing mass. Beside being of independent interest, our alternative proof suggests a natural approach to strengthen, and expand, the recent results on the rate of convergence of the Good-Turing estimator under α ∈ (0 , 1) regularly varying type’s proportions. In particular, we show that the convergence rate n − α/ 2 is the best rate that any estimator can achieve, up to a slowly varying function. Furthermore, we prove that a lower bound to the minimax estimation risk must scale at least as n − α/ 2 , which leads to conjecture that the Good-Turing estimator is a rate optimal minimax estimator under regularly varying type proportions.
{"title":"On consistent and rate optimal estimation of the missing mass","authors":"Fadhel Ayed, M. Battiston, F. Camerlenghi, S. Favaro","doi":"10.1214/20-AIHP1126","DOIUrl":"https://doi.org/10.1214/20-AIHP1126","url":null,"abstract":". Given n samples from a population of individuals belonging to different types with unknown proportions, how do we estimate the probability of discovering a new type at the ( n + 1)-th draw? This is a classical problem in statistics, commonly referred to as the missing mass estimation problem. Recent results have shown: i) the impossibility of estimating the missing mass without imposing further assumptions on type’s proportions; ii) the consistency of the Good-Turing estimator of the missing mass under the assumption that the tail of type’s proportions decays to zero as a regularly varying function with parameter α ∈ (0 , 1); ii) the rate of convergence n − α/ 2 for the Good-Turing estimator under the class of α ∈ (0 , 1) regularly varying P . In this paper we introduce an alternative, and remarkably shorter, proof of the impossibility of a distribution-free estimation of the missing mass. Beside being of independent interest, our alternative proof suggests a natural approach to strengthen, and expand, the recent results on the rate of convergence of the Good-Turing estimator under α ∈ (0 , 1) regularly varying type’s proportions. In particular, we show that the convergence rate n − α/ 2 is the best rate that any estimator can achieve, up to a slowly varying function. Furthermore, we prove that a lower bound to the minimax estimation risk must scale at least as n − α/ 2 , which leads to conjecture that the Good-Turing estimator is a rate optimal minimax estimator under regularly varying type proportions.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"51 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82047308","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}
Abstract: In the density estimation model, the question of adaptive inference using Pólya tree–type prior distributions is considered. A class of prior densities having a tree structure, called spike–and–slab Pólya trees, is introduced. For this class, two types of results are obtained: first, the Bayesian posterior distribution is shown to converge at the minimax rate for the supremum norm in an adaptive way, for any Hölder regularity of the true density between 0 and 1, thereby providing adaptive counterparts to the results for classical Pólya trees in [5]. Second, the question of uncertainty quantification is considered. An adaptive nonparametric Bernstein– von Mises theorem is derived. Next, it is shown that, under a self-similarity condition on the true density, certain credible sets from the posterior distribution are adaptive confidence bands, having prescribed coverage level and with a diameter shrinking at optimal rate in the minimax sense.
摘要:在密度估计模型中，考虑了利用Pólya树型先验分布进行自适应推理的问题。介绍了一类具有树形结构的先验密度，称为spike-and-slab Pólya树。对于这一类，得到了两类结果:第一，对于0和1之间的任何Hölder真密度的正则性，贝叶斯后验分布以自适应的方式收敛于最大范数的极小极大率，从而提供了[5]中经典Pólya树的结果的自适应对应。其次，考虑了不确定度的量化问题。导出了一个自适应非参数Bernstein - von Mises定理。其次，在真密度的自相似条件下，来自后验分布的某些可信集是自适应置信带，具有规定的覆盖水平，直径在极小极大意义上以最优速率收缩。
{"title":"Spike and slab Pólya tree posterior densities: Adaptive inference","authors":"I. Castillo, Romain Mismer","doi":"10.1214/20-AIHP1132","DOIUrl":"https://doi.org/10.1214/20-AIHP1132","url":null,"abstract":"Abstract: In the density estimation model, the question of adaptive inference using Pólya tree–type prior distributions is considered. A class of prior densities having a tree structure, called spike–and–slab Pólya trees, is introduced. For this class, two types of results are obtained: first, the Bayesian posterior distribution is shown to converge at the minimax rate for the supremum norm in an adaptive way, for any Hölder regularity of the true density between 0 and 1, thereby providing adaptive counterparts to the results for classical Pólya trees in [5]. Second, the question of uncertainty quantification is considered. An adaptive nonparametric Bernstein– von Mises theorem is derived. Next, it is shown that, under a self-similarity condition on the true density, certain credible sets from the posterior distribution are adaptive confidence bands, having prescribed coverage level and with a diameter shrinking at optimal rate in the minimax sense.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"39 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76607013","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}
Zhan Shi, V. Sidoravicius, He Song, Longmin Wang, Kainan Xiang
The uniform spanning forest measure (USF) on a locally finite, infinite connected graph with conductance c is defined as a weak limit of uniform spanning tree measure on finite subgraphs. Depending on the underlying graph and conductances, the corresponding USF is not necessarily concentrated on the set of spanning trees. Pemantle [20] showed that on Z, equipped with the unit conductance c = 1, USF is concentrated on spanning trees if and only if d ≤ 4. In this work we study the USF associated with conductances c(e) = λ−|e|, where |e| is the graph distance of the edge e from the origin, and λ ∈ (0, 1) is a fixed parameter. Our main result states that in this case USF consists of finitely many trees if and only if d = 2 or 3. More precisely, we prove that the uniform spanning forest has 2 trees if d = 2 or 3, and infinitely many trees if d ≥ 4. Our method relies on the analysis of the spectral radius and the speed of the λ-biased random walk on Z. AMS 2010 subject classifications. Primary 60J10, 60G50, 05C81; secondary 60C05, 05C63, 05C80.
{"title":"Uniform spanning forests on biased Euclidean lattices","authors":"Zhan Shi, V. Sidoravicius, He Song, Longmin Wang, Kainan Xiang","doi":"10.1214/20-AIHP1119","DOIUrl":"https://doi.org/10.1214/20-AIHP1119","url":null,"abstract":"The uniform spanning forest measure (USF) on a locally finite, infinite connected graph with conductance c is defined as a weak limit of uniform spanning tree measure on finite subgraphs. Depending on the underlying graph and conductances, the corresponding USF is not necessarily concentrated on the set of spanning trees. Pemantle [20] showed that on Z, equipped with the unit conductance c = 1, USF is concentrated on spanning trees if and only if d ≤ 4. In this work we study the USF associated with conductances c(e) = λ−|e|, where |e| is the graph distance of the edge e from the origin, and λ ∈ (0, 1) is a fixed parameter. Our main result states that in this case USF consists of finitely many trees if and only if d = 2 or 3. More precisely, we prove that the uniform spanning forest has 2 trees if d = 2 or 3, and infinitely many trees if d ≥ 4. Our method relies on the analysis of the spectral radius and the speed of the λ-biased random walk on Z. AMS 2010 subject classifications. Primary 60J10, 60G50, 05C81; secondary 60C05, 05C63, 05C80.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"46 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83599600","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}
D. Belomestny, Vytaut.e Pilipauskait.e, M. Podolskij
In this paper we study the problem of semiparametric estimation for a class of McKean-Vlasov stochastic differential equations. Our aim is to estimate the drift coefficient of a MV-SDE based on observations of the corresponding particle system. We propose a semiparametric estimation procedure and derive the rates of convergence for the resulting estimator. We further prove that the obtained rates are essentially optimal in the minimax sense.
{"title":"Semiparametric estimation of McKean–Vlasov SDEs","authors":"D. Belomestny, Vytaut.e Pilipauskait.e, M. Podolskij","doi":"10.1214/22-aihp1261","DOIUrl":"https://doi.org/10.1214/22-aihp1261","url":null,"abstract":"In this paper we study the problem of semiparametric estimation for a class of McKean-Vlasov stochastic differential equations. Our aim is to estimate the drift coefficient of a MV-SDE based on observations of the corresponding particle system. We propose a semiparametric estimation procedure and derive the rates of convergence for the resulting estimator. We further prove that the obtained rates are essentially optimal in the minimax sense.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"120 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86156172","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 introduce"$t$-LC triangulated manifolds"as those triangulations obtainable from a tree of $d$-simplices by recursively identifying two boundary $(d-1)$-faces whose intersection has dimension at least $d-t-1$. The $t$-LC notion interpolates between the class of LC manifolds introduced by Durhuus--Jonsson (corresponding to the case $t=1$), and the class of all manifolds (case $t=d$). Benedetti--Ziegler proved that there are at most $2^{d^2 , N}$ triangulated $1$-LC $d$-manifolds with $N$ facets. Here we prove that there are at most $2^{frac{d^3}{2}N}$ triangulated $2$-LC $d$-manifolds with $N$ facets. This extends to all dimensions an intuition by Mogami for $d=3$. We also introduce"$t$-constructible complexes", interpolating between constructible complexes (the case $t=1$) and all complexes (case $t=d$). We show that all $t$-constructible pseudomanifolds are $t$-LC, and that all $t$-constructible complexes have (homotopical) depth larger than $d-t$. This extends the famous result by Hochster that constructible complexes are (homotopy) Cohen--Macaulay.
{"title":"2-LC triangulated manifolds are exponentially many","authors":"Bruno Benedetti, Marta Pavelka","doi":"10.4171/aihpd/170","DOIUrl":"https://doi.org/10.4171/aihpd/170","url":null,"abstract":"We introduce\"$t$-LC triangulated manifolds\"as those triangulations obtainable from a tree of $d$-simplices by recursively identifying two boundary $(d-1)$-faces whose intersection has dimension at least $d-t-1$. The $t$-LC notion interpolates between the class of LC manifolds introduced by Durhuus--Jonsson (corresponding to the case $t=1$), and the class of all manifolds (case $t=d$). Benedetti--Ziegler proved that there are at most $2^{d^2 , N}$ triangulated $1$-LC $d$-manifolds with $N$ facets. Here we prove that there are at most $2^{frac{d^3}{2}N}$ triangulated $2$-LC $d$-manifolds with $N$ facets. This extends to all dimensions an intuition by Mogami for $d=3$. We also introduce\"$t$-constructible complexes\", interpolating between constructible complexes (the case $t=1$) and all complexes (case $t=d$). We show that all $t$-constructible pseudomanifolds are $t$-LC, and that all $t$-constructible complexes have (homotopical) depth larger than $d-t$. This extends the famous result by Hochster that constructible complexes are (homotopy) Cohen--Macaulay.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"20 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77643832","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}
It is shown that when $dge 3$, the growing random surface generated by the $(d+1)$-dimensional directed polymer model at sufficiently high temperature, after being smoothed by taking microscopic local averages, converges to a solution of the deterministic KPZ equation in a suitable scaling limit.
{"title":"Weak convergence of directed polymers to deterministic KPZ at high temperature","authors":"S. Chatterjee","doi":"10.1214/22-aihp1287","DOIUrl":"https://doi.org/10.1214/22-aihp1287","url":null,"abstract":"It is shown that when $dge 3$, the growing random surface generated by the $(d+1)$-dimensional directed polymer model at sufficiently high temperature, after being smoothed by taking microscopic local averages, converges to a solution of the deterministic KPZ equation in a suitable scaling limit.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"33 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81306973","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 deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modeled by considering perturbations of the transition probabilities within a proximity in Wasserstein distance. As a limit over progressively finer time periods, on which the level of uncertainty scales proportionally, we obtain a convex semigroup satisfying a nonlinear PDE in a viscosity sense. A remarkable observation is that, in standard situations, the nonlinear transition operators arising from nonparametric uncertainty coincide with the ones related to parametric drift uncertainty. On the level of the generator, the uncertainty is reflected as an additive perturbation in terms of a convex functional of first order derivatives. We additionally provide sensitivity bounds for the convex semigroup relative to the reference model. The results are illustrated with Wasserstein perturbations of L'evy processes, infinite-dimensional Ornstein-Uhlenbeck processes, geometric Brownian motions, and Koopman semigroups.
{"title":"Wasserstein perturbations of Markovian transition semigroups","authors":"Sven Fuhrmann, M. Kupper, M. Nendel","doi":"10.1214/22-aihp1270","DOIUrl":"https://doi.org/10.1214/22-aihp1270","url":null,"abstract":"In this paper, we deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modeled by considering perturbations of the transition probabilities within a proximity in Wasserstein distance. As a limit over progressively finer time periods, on which the level of uncertainty scales proportionally, we obtain a convex semigroup satisfying a nonlinear PDE in a viscosity sense. A remarkable observation is that, in standard situations, the nonlinear transition operators arising from nonparametric uncertainty coincide with the ones related to parametric drift uncertainty. On the level of the generator, the uncertainty is reflected as an additive perturbation in terms of a convex functional of first order derivatives. We additionally provide sensitivity bounds for the convex semigroup relative to the reference model. The results are illustrated with Wasserstein perturbations of L'evy processes, infinite-dimensional Ornstein-Uhlenbeck processes, geometric Brownian motions, and Koopman semigroups.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"37 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86565702","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}
For i.i.d. random vectors $(M_{1},Q_{1}),(M_{2},Q_{2}),ldots$ such that $M>0$ a.s., $Qgeq 0$ a.s. and $mathbb{P}(Q=0)<1$, the random difference equation $X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,ldots$, is studied in the critical case when the random walk with increments $log M_{1},log M_{2}$ is oscillating. We provide conditions for the null-recurrence and transience of the Markov chain $(X_{n})_{nge 0}$ by inter alia drawing on techniques developed in the related article Alsmeyer et al (2017) for another case exhibiting the null-recurrence/transience dichotomy.
{"title":"Recurrence and transience of random difference equations in the critical case","authors":"G. Alsmeyer, A. Iksanov","doi":"10.1214/22-aihp1274","DOIUrl":"https://doi.org/10.1214/22-aihp1274","url":null,"abstract":"For i.i.d. random vectors $(M_{1},Q_{1}),(M_{2},Q_{2}),ldots$ such that $M>0$ a.s., $Qgeq 0$ a.s. and $mathbb{P}(Q=0)<1$, the random difference equation $X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,ldots$, is studied in the critical case when the random walk with increments $log M_{1},log M_{2}$ is oscillating. We provide conditions for the null-recurrence and transience of the Markov chain $(X_{n})_{nge 0}$ by inter alia drawing on techniques developed in the related article Alsmeyer et al (2017) for another case exhibiting the null-recurrence/transience dichotomy.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"46 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72534167","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}
Let $P$ be a bistochastic matrix of size $n$, and let $Pi$ be a permutation matrix of size $n$. In this paper, we are interested in the mixing time of the Markov chain whose transition matrix is given by $Q=PPi$. In other words, the chain alternates between random steps governed by $P$ and deterministic steps governed by $Pi$. We show that if the permutation $Pi$ is chosen uniformly at random, then under mild assumptions on $P$, with high probability, the chain $Q$ exhibits cutoff at time $frac{log n}{mathbf{h}}$, where $mathbf{h}$ is the entropic rate of $P$. Moreover, for deterministic permutations, we improve the upper bound on the mixing time obtained by Chatterjee and Diaconis (2020).
{"title":"Cutoff for permuted Markov chains","authors":"Anna Ben-Hamou, Y. Peres","doi":"10.1214/22-aihp1248","DOIUrl":"https://doi.org/10.1214/22-aihp1248","url":null,"abstract":"Let $P$ be a bistochastic matrix of size $n$, and let $Pi$ be a permutation matrix of size $n$. In this paper, we are interested in the mixing time of the Markov chain whose transition matrix is given by $Q=PPi$. In other words, the chain alternates between random steps governed by $P$ and deterministic steps governed by $Pi$. We show that if the permutation $Pi$ is chosen uniformly at random, then under mild assumptions on $P$, with high probability, the chain $Q$ exhibits cutoff at time $frac{log n}{mathbf{h}}$, where $mathbf{h}$ is the entropic rate of $P$. Moreover, for deterministic permutations, we improve the upper bound on the mixing time obtained by Chatterjee and Diaconis (2020).","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"84 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77501990","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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