We consider the standard first passage percolation model on Z^d with a distribution G on R+ that admits an exponential moment. We study the maximal flow between a compact convex subset A of R^d and infinity. The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut A from infinity. We prove that the rescaled maximal flow between nA and infinity φ(nA)/n^ (d−1) almost surely converges towards a deterministic constant depending on A. This constant corresponds to the capacity of the boundary ∂A of A and is the integral of a deterministic function over ∂A. This result was shown in dimension 2 and conjectured for higher dimensions by Garet in [6].
{"title":"The maximal flow from a compact convex subset to infinity in first passage percolation on $mathbb{Z}^{d}$","authors":"Barbara Dembin","doi":"10.1214/19-aop1367","DOIUrl":"https://doi.org/10.1214/19-aop1367","url":null,"abstract":"We consider the standard first passage percolation model on Z^d with a distribution G on R+ that admits an exponential moment. We study the maximal flow between a compact convex subset A of R^d and infinity. The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut A from infinity. We prove that the rescaled maximal flow between nA and infinity φ(nA)/n^ (d−1) almost surely converges towards a deterministic constant depending on A. This constant corresponds to the capacity of the boundary ∂A of A and is the integral of a deterministic function over ∂A. This result was shown in dimension 2 and conjectured for higher dimensions by Garet in [6].","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"48 1","pages":"622-645"},"PeriodicalIF":2.3,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47930900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients","authors":"M. Hutzenthaler, Arnulf Jentzen","doi":"10.1214/19-aop1345","DOIUrl":"https://doi.org/10.1214/19-aop1345","url":null,"abstract":"","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"48 1","pages":"53-93"},"PeriodicalIF":2.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66077577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-01Epub Date: 2019-11-12DOI: 10.1007/s00262-019-02429-2
Yangyang Qi, Yuan Chang, Zewei Wang, Lingli Chen, Yunyi Kong, Peipei Zhang, Zheng Liu, Quan Zhou, Yifan Chen, Jiajun Wang, Qi Bai, Yu Xia, Li Liu, Yu Zhu, Le Xu, Bo Dai, Jianming Guo, Yiwei Wang, Jiejie Xu, Weijuan Zhang
Purpose: Tumor-associated macrophages (TAMs) exist as heterogeneous subsets and have dichotomous roles in cancer-immune evasion. This study aims to assess the clinical effects of Galectin-9+ tumor-associated macrophages (Gal-9+TAMs) in muscle-invasive bladder cancer (MIBC).
Experimental design: We identified Gal-9+TAMs by immunohistochemistry (IHC) analysis of a tumor microarray (TMA) (n = 141) from the Zhongshan Hospital and by flow cytometric analysis of tumor specimens (n = 20) from the Shanghai Cancer Center. The survival benefit of platinum-based chemotherapy in this subpopulation was evaluated. The effect of the tumor-immune microenvironment with different percentages of Gal-9+TAMs was explored.
Results: The frequency of Gal-9+TAMs increased with tumor stage and grade. Gal-9+TAMs predicted poor overall survival (OS) and recurrence-free survival (RFS) and were better than Gal-9-TAMs and TAMs to discriminate prognostic groups. In univariate and multivariate Cox regression analyses, patients with high percentages of Gal-9+TAMs showed the prominent survival benefit after receiving adjuvant chemotherapy (ACT). High Gal-9+TAM infiltration correlated with increasing numbers of regulatory T cells (Tregs) and mast cells and decreasing numbers of CD8+T and dendritic cells (DCs). Dense infiltration of Gal-9+TAMs was related to reduced cytotoxic molecules, enhanced immune checkpoints or immunosuppressive cytokines expressed by immune cells, as well as active proliferation of tumor cells. Additionally, the subpopulation accumulated was strongly associated with PD-1+TIM-3+CD8+T cells.
Conclusions: Gal-9+TAMs predicted OS and RFS and response to ACT in MIBC patients. High Gal-9+TAMs were associated with a pro-tumor immune contexture concomitant with T cell exhaustion.
{"title":"Tumor-associated macrophages expressing galectin-9 identify immunoevasive subtype muscle-invasive bladder cancer with poor prognosis but favorable adjuvant chemotherapeutic response.","authors":"Yangyang Qi, Yuan Chang, Zewei Wang, Lingli Chen, Yunyi Kong, Peipei Zhang, Zheng Liu, Quan Zhou, Yifan Chen, Jiajun Wang, Qi Bai, Yu Xia, Li Liu, Yu Zhu, Le Xu, Bo Dai, Jianming Guo, Yiwei Wang, Jiejie Xu, Weijuan Zhang","doi":"10.1007/s00262-019-02429-2","DOIUrl":"10.1007/s00262-019-02429-2","url":null,"abstract":"<p><strong>Purpose: </strong>Tumor-associated macrophages (TAMs) exist as heterogeneous subsets and have dichotomous roles in cancer-immune evasion. This study aims to assess the clinical effects of Galectin-9<sup>+</sup> tumor-associated macrophages (Gal-9<sup>+</sup>TAMs) in muscle-invasive bladder cancer (MIBC).</p><p><strong>Experimental design: </strong>We identified Gal-9<sup>+</sup>TAMs by immunohistochemistry (IHC) analysis of a tumor microarray (TMA) (n = 141) from the Zhongshan Hospital and by flow cytometric analysis of tumor specimens (n = 20) from the Shanghai Cancer Center. The survival benefit of platinum-based chemotherapy in this subpopulation was evaluated. The effect of the tumor-immune microenvironment with different percentages of Gal-9<sup>+</sup>TAMs was explored.</p><p><strong>Results: </strong>The frequency of Gal-9<sup>+</sup>TAMs increased with tumor stage and grade. Gal-9<sup>+</sup>TAMs predicted poor overall survival (OS) and recurrence-free survival (RFS) and were better than Gal-9<sup>-</sup>TAMs and TAMs to discriminate prognostic groups. In univariate and multivariate Cox regression analyses, patients with high percentages of Gal-9<sup>+</sup>TAMs showed the prominent survival benefit after receiving adjuvant chemotherapy (ACT). High Gal-9<sup>+</sup>TAM infiltration correlated with increasing numbers of regulatory T cells (Tregs) and mast cells and decreasing numbers of CD8<sup>+</sup>T and dendritic cells (DCs). Dense infiltration of Gal-9<sup>+</sup>TAMs was related to reduced cytotoxic molecules, enhanced immune checkpoints or immunosuppressive cytokines expressed by immune cells, as well as active proliferation of tumor cells. Additionally, the subpopulation accumulated was strongly associated with PD-1<sup>+</sup>TIM-3<sup>+</sup>CD8<sup>+</sup>T cells.</p><p><strong>Conclusions: </strong>Gal-9<sup>+</sup>TAMs predicted OS and RFS and response to ACT in MIBC patients. High Gal-9<sup>+</sup>TAMs were associated with a pro-tumor immune contexture concomitant with T cell exhaustion.</p>","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"4 1","pages":"2067-2080"},"PeriodicalIF":5.8,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11028176/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88323273","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a ballistic strong law of large numbers and an invariance principle for random walks in strong mixing environments, under condition (T ) of Sznitman (cf. [Sz01]). This weakens for the first time Kalikow’s ballisticity assumption on mixing environments and proves the existence of arbitrary finite order moments for the approximate regeneration time of F. Comets and O. Zeitouni [CZ02]. The main technical tool in the proof is the introduction of renormalization schemes, which had only been considered for i.i.d. environments.
{"title":"On the transient (T) condition for random walk in mixing environment","authors":"E. Aguilar","doi":"10.1214/18-AOP1330","DOIUrl":"https://doi.org/10.1214/18-AOP1330","url":null,"abstract":"We prove a ballistic strong law of large numbers and an invariance principle for random walks in strong mixing environments, under condition (T ) of Sznitman (cf. [Sz01]). This weakens for the first time Kalikow’s ballisticity assumption on mixing environments and proves the existence of arbitrary finite order moments for the approximate regeneration time of F. Comets and O. Zeitouni [CZ02]. The main technical tool in the proof is the introduction of renormalization schemes, which had only been considered for i.i.d. environments.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"49 1","pages":"3003-3054"},"PeriodicalIF":2.3,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77094245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For d≥2d≥2 and n∈Nn∈N even, let pn=pn(d)pn=pn(d) denote the number of length nn self-avoiding polygons in ZdZd up to translation. The polygon cardinality grows exponentially, and the growth rate limn∈2Np1/nn∈(0,∞)limn∈2Npn1/n∈(0,∞) is called the connective constant and denoted by μμ. Madras [J. Stat. Phys. 78 (1995) 681–699] has shown that pnμ−n≤Cn−1/2pnμ−n≤Cn−1/2 in dimension d=2d=2. Here, we establish that pnμ−n≤n−3/2+o(1)pnμ−n≤n−3/2+o(1) for a set of even nn of full density when d=2d=2. We also consider a certain variant of self-avoiding walk and argue that, when d≥3d≥3, an upper bound of n−2+d−1+o(1)n−2+d−1+o(1) holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.
{"title":"An upper bound on the number of self-avoiding polygons via joining","authors":"A. Hammond","doi":"10.1214/17-AOP1182","DOIUrl":"https://doi.org/10.1214/17-AOP1182","url":null,"abstract":"For d≥2d≥2 and n∈Nn∈N even, let pn=pn(d)pn=pn(d) denote the number of length nn self-avoiding polygons in ZdZd up to translation. The polygon cardinality grows exponentially, and the growth rate limn∈2Np1/nn∈(0,∞)limn∈2Npn1/n∈(0,∞) is called the connective constant and denoted by μμ. Madras [J. Stat. Phys. 78 (1995) 681–699] has shown that pnμ−n≤Cn−1/2pnμ−n≤Cn−1/2 in dimension d=2d=2. Here, we establish that pnμ−n≤n−3/2+o(1)pnμ−n≤n−3/2+o(1) for a set of even nn of full density when d=2d=2. We also consider a certain variant of self-avoiding walk and argue that, when d≥3d≥3, an upper bound of n−2+d−1+o(1)n−2+d−1+o(1) holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"46 1","pages":"175-206"},"PeriodicalIF":2.3,"publicationDate":"2018-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOP1182","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48814709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In [Ald00], Aldous investigates a symmetric Markov chain on cladograms and gives bounds on its mixing and relaxation times. The latter bound was sharpened in [Sch02]. In the present paper we encode cladograms as binary, algebraic measure trees and show that this Markov chain on cladograms with fixed number of leaves converges in distribution as the number of leaves goes to infinity. We give a rigorous construction of the limit, whose existence was conjectured by Aldous and which we therefore refer to as Aldous diffusion, as a solution of a well-posed martingale problem. We show that the Aldous diffusion is a Feller process with continuous paths, and the algebraic measure Brownian CRT is its unique invariant distribution. Furthermore, we consider the vector of the masses of the three subtrees connected to a sampled branch point. In the Brownian CRT, its annealed law is known to be the Dirichlet distribution. Here, we give an explicit expression for the infinitesimal evolution of its quenched law under the Aldous diffusion.
{"title":"The Aldous chain on cladograms in the diffusion limit","authors":"Wolfgang Lohr, L. Mytnik, A. Winter","doi":"10.1214/20-AOP1431","DOIUrl":"https://doi.org/10.1214/20-AOP1431","url":null,"abstract":"In [Ald00], Aldous investigates a symmetric Markov chain on cladograms and gives bounds on its mixing and relaxation times. The latter bound was sharpened in [Sch02]. In the present paper we encode cladograms as binary, algebraic measure trees and show that this Markov chain on cladograms with fixed number of leaves converges in distribution as the number of leaves goes to infinity. We give a rigorous construction of the limit, whose existence was conjectured by Aldous and which we therefore refer to as Aldous diffusion, as a solution of a well-posed martingale problem. We show that the Aldous diffusion is a Feller process with continuous paths, and the algebraic measure Brownian CRT is its unique invariant distribution. Furthermore, we consider the vector of the masses of the three subtrees connected to a sampled branch point. In the Brownian CRT, its annealed law is known to be the Dirichlet distribution. Here, we give an explicit expression for the infinitesimal evolution of its quenched law under the Aldous diffusion.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"48 1","pages":"2565-2590"},"PeriodicalIF":2.3,"publicationDate":"2018-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46622993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Benson Au, Guillaume C'ebron, Antoine Dahlqvist, Franck Gabriel, C. Male
We prove that independent families of permutation invariant random matrices are asymptotically free over the diagonal, both in probability and in expectation, under a uniform boundedness assumption on the operator norm. We can relax the operator norm assumption to an estimate on sums associated to graphs of matrices, further extending the range of applications (for example, to Wigner matrices with exploding moments and so the sparse regime of the Erdős-Renyi model). The result still holds even if the matrices are multiplied entrywise by bounded random variables (for example, as in the case of matrices with a variance profile and percolation models).
{"title":"Freeness over the diagonal for large random matrices","authors":"Benson Au, Guillaume C'ebron, Antoine Dahlqvist, Franck Gabriel, C. Male","doi":"10.1214/20-AOP1447","DOIUrl":"https://doi.org/10.1214/20-AOP1447","url":null,"abstract":"We prove that independent families of permutation invariant random matrices are asymptotically free over the diagonal, both in probability and in expectation, under a uniform boundedness assumption on the operator norm. We can relax the operator norm assumption to an estimate on sums associated to graphs of matrices, further extending the range of applications (for example, to Wigner matrices with exploding moments and so the sparse regime of the Erdős-Renyi model). The result still holds even if the matrices are multiplied entrywise by bounded random variables (for example, as in the case of matrices with a variance profile and percolation models).","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"1 1","pages":""},"PeriodicalIF":2.3,"publicationDate":"2018-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66080625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A nonbacktracking walk on a graph is a directed path such that no edge is the inverse of its preceding edge. The nonbacktracking matrix of a graph is indexed by its directed edges and can be used to count nonbacktracking walks of a given length. It has been used recently in the context of community detection and has appeared previously in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. In this work, we study the largest eigenvalues of the nonbacktracking matrix of the Erdős–Renyi random graph and of the stochastic block model in the regime where the number of edges is proportional to the number of vertices. Our results confirm the “spectral redemption conjecture” in the symmetric case and show that community detection can be made on the basis of the leading eigenvectors above the feasibility threshold.
{"title":"Nonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphs","authors":"C. Bordenave, M. Lelarge, L. Massoulié","doi":"10.1214/16-AOP1142","DOIUrl":"https://doi.org/10.1214/16-AOP1142","url":null,"abstract":"A nonbacktracking walk on a graph is a directed path such that no edge is the inverse of its preceding edge. The nonbacktracking matrix of a graph is indexed by its directed edges and can be used to count nonbacktracking walks of a given length. It has been used recently in the context of community detection and has appeared previously in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. In this work, we study the largest eigenvalues of the nonbacktracking matrix of the Erdős–Renyi random graph and of the stochastic block model in the regime where the number of edges is proportional to the number of vertices. Our results confirm the “spectral redemption conjecture” in the symmetric case and show that community detection can be made on the basis of the leading eigenvectors above the feasibility threshold.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"46 1","pages":"1-71"},"PeriodicalIF":2.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1142","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Existence conditions of permanental distributions are deeply connected to existence conditions of multivariate negative binomial distributions. The aim of this paper is twofold. It answers several questions generated by recent works on this subject, but it also goes back to the roots of this field and fixes existing gaps in older papers concerning conditions of infinite divisibility for these distributions.
{"title":"Existence conditions of permanental and multivariate negative binomial distributions","authors":"Nathalie Eisenbaum, F. Maunoury","doi":"10.1214/17-AOP1179","DOIUrl":"https://doi.org/10.1214/17-AOP1179","url":null,"abstract":"Existence conditions of permanental distributions are deeply connected to existence conditions of multivariate negative binomial distributions. The aim of this paper is twofold. It answers several questions generated by recent works on this subject, but it also goes back to the roots of this field and fixes existing gaps in older papers concerning conditions of infinite divisibility for these distributions.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"4786-4820"},"PeriodicalIF":2.3,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOP1179","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42298657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present some new limit theorems for power variation of kkth order increments of stationary increments Levy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives novel results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we show that the first-order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments k≥1k≥1, the considered power p>0p>0, the Blumenthal–Getoor index β∈[0,2)β∈[0,2) of the driving pure jump Levy process LL and the behaviour of the kernel function gg at 00 determined by the power αα. First-order asymptotic theory essentially comprises three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove a second-order limit theorem connected to the ergodic type result. When the driving Levy process LL is a symmetric ββ-stable process, we obtain two different limits: a central limit theorem and convergence in distribution towards a (k−α)β(k−α)β-stable totally right skewed random variable.
{"title":"Power variation for a class of stationary increments Lévy driven moving averages","authors":"A. Basse-O’Connor, R. Lachièze-Rey, M. Podolskij","doi":"10.1214/16-AOP1170","DOIUrl":"https://doi.org/10.1214/16-AOP1170","url":null,"abstract":"In this paper, we present some new limit theorems for power variation of kkth order increments of stationary increments Levy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives novel results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we show that the first-order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments k≥1k≥1, the considered power p>0p>0, the Blumenthal–Getoor index β∈[0,2)β∈[0,2) of the driving pure jump Levy process LL and the behaviour of the kernel function gg at 00 determined by the power αα. First-order asymptotic theory essentially comprises three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove a second-order limit theorem connected to the ergodic type result. When the driving Levy process LL is a symmetric ββ-stable process, we obtain two different limits: a central limit theorem and convergence in distribution towards a (k−α)β(k−α)β-stable totally right skewed random variable.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"4477-4528"},"PeriodicalIF":2.3,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1170","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45244732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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