We augment standard branching Brownian motion by adding a competitive interaction between nearby particles. Informally, when particles are in competition, the local resources are insufficient to cover the energetic cost of motion, so the particles’ masses decay. In standard BBM, we may define the front displacement at time tt as the greatest distance of a particle from the origin. For the model with masses, it makes sense to instead define the front displacement as the distance at which the local mass density drops from Θ(1)Θ(1) to o(1)o(1). We show that one can find arbitrarily large times tt for which this occurs at a distance Θ(t1/3)Θ(t1/3) behind the front displacement for standard BBM.
{"title":"The front location in branching Brownian motion with decay of mass","authors":"L. Addario-Berry, S. Penington","doi":"10.1214/16-AOP1148","DOIUrl":"https://doi.org/10.1214/16-AOP1148","url":null,"abstract":"We augment standard branching Brownian motion by adding a competitive interaction between nearby particles. Informally, when particles are in competition, the local resources are insufficient to cover the energetic cost of motion, so the particles’ masses decay. In standard BBM, we may define the front displacement at time tt as the greatest distance of a particle from the origin. For the model with masses, it makes sense to instead define the front displacement as the distance at which the local mass density drops from Θ(1)Θ(1) to o(1)o(1). We show that one can find arbitrarily large times tt for which this occurs at a distance Θ(t1/3)Θ(t1/3) behind the front displacement for standard BBM.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"3752-3794"},"PeriodicalIF":2.3,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1148","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46509277","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}
We provide a Clark–Ocone formula for square-integrable functionals of a general temporal point process satisfying only a mild moment condition, generalizing known results on the Poisson space. Some classical applications are given, namely a deviation bound and the construction of a hedging portfolio in a pure-jump market model. As a more modern application, we provide a bound on the total variation distance between two temporal point processes, improving in some sense a recent result in this direction.
{"title":"A Clark–Ocone formula for temporal point processes and applications","authors":"I. Flint, G. Torrisi","doi":"10.1214/16-AOP1136","DOIUrl":"https://doi.org/10.1214/16-AOP1136","url":null,"abstract":"We provide a Clark–Ocone formula for square-integrable functionals of a general temporal point process satisfying only a mild moment condition, generalizing known results on the Poisson space. Some classical applications are given, namely a deviation bound and the construction of a hedging portfolio in a pure-jump market model. As a more modern application, we provide a bound on the total variation distance between two temporal point processes, improving in some sense a recent result in this direction.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"3266-3292"},"PeriodicalIF":2.3,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1136","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47283691","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}
We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in population genetics, the so-called generalized Kimura diffusion operators Epstein and Mazzeo [SIAM J. Math. Anal. 42 (2010) 568–608; Degenerate Diffusion Operators Arising in Population Biology (2013) Princeton University Press; Applied Mathematics Research Express (2016)]. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients and the proof of the scale-invariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on Feynman–Kac formulas concerning the assumptions on the degeneracy of the diffusion matrix, the boundedness of the drift coefficients and the a priori regularity of the weak solutions.
本文研究了种群遗传学中出现的一类退化扩散算子的各种概率和解析性质，即广义Kimura扩散算子Epstein和Mazzeo [SIAM J. Math]。肛门42 (2010)568-608;种群生物学中的退化扩散算子(2013)普林斯顿大学出版社;应用数学研究快报(2016)。我们的主要成果是退化低阶系数奇异抛物方程弱解的随机表示和Kimura抛物方程非负解的尺度不变Harnack不等式的证明。我们所建立的解的随机表示是对经典费曼-卡茨公式关于扩散矩阵的简并性、漂移系数的有界性和弱解的先验正则性等假设的结果的一个相当大的推广。
{"title":"The Feynman–Kac formula and Harnack inequality for degenerate diffusions","authors":"C. Epstein, C. Pop","doi":"10.1214/16-AOP1138","DOIUrl":"https://doi.org/10.1214/16-AOP1138","url":null,"abstract":"We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in population genetics, the so-called generalized Kimura diffusion operators Epstein and Mazzeo [SIAM J. Math. Anal. 42 (2010) 568–608; Degenerate Diffusion Operators Arising in Population Biology (2013) Princeton University Press; Applied Mathematics Research Express (2016)]. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients and the proof of the scale-invariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on Feynman–Kac formulas concerning the assumptions on the degeneracy of the diffusion matrix, the boundedness of the drift coefficients and the a priori regularity of the weak solutions.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"3336-3384"},"PeriodicalIF":2.3,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1138","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46532803","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 permanental vector with a symmetric kernel and index 22 is a squared Gaussian vector. The definition of permanental vectors is a natural extension of the definition of squared Gaussian vectors to nonsymmetric kernels and to positive indexes. The only known permanental vectors either have a positive definite kernel or are infinitely divisible. Are there some others? We present a partial answer to this question.
{"title":"Permanental vectors with nonsymmetric kernels","authors":"Nathalie Eisenbaum","doi":"10.1214/15-AOP1008","DOIUrl":"https://doi.org/10.1214/15-AOP1008","url":null,"abstract":"A permanental vector with a symmetric kernel and index 22 is a squared Gaussian vector. The definition of permanental vectors is a natural extension of the definition of squared Gaussian vectors to nonsymmetric kernels and to positive indexes. The only known permanental vectors either have a positive definite kernel or are infinitely divisible. Are there some others? We present a partial answer to this question.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"210-224"},"PeriodicalIF":2.3,"publicationDate":"2017-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49029704","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}
We prove a functional central limit theorem for partial sums of symmetric stationary long-range dependent heavy tailed infinitely divisible processes. The limiting stable process is particularly interesting due to its long memory which is quantified by a Mittag–Leffler process induced by an associated Harris chain, at the discrete-time level. Previous results in Owada and Samorodnitsky [Ann. Probab. 43 (2015) 240–285] dealt with positive dependence in the increment process, whereas this paper derives the functional limit theorems under negative dependence. The negative dependence is due to cancellations arising from Gaussian-type fluctuations of functionals of the associated Harris chain. The new types of limiting processes involve stable random measures, due to heavy tails, Mittag–Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations. Along the way, we prove a function central limit theorem for fluctuations of functionals of Harris chains which is of independent interest as it extends a result of Chen [Probab. Theory Related Fields 116 (2000) 89–123].
{"title":"Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows","authors":"Paul Jung, Takashi Owada, G. Samorodnitsky","doi":"10.1214/16-AOP1107","DOIUrl":"https://doi.org/10.1214/16-AOP1107","url":null,"abstract":"We prove a functional central limit theorem for partial sums of symmetric stationary long-range dependent heavy tailed infinitely divisible processes. The limiting stable process is particularly interesting due to its long memory which is quantified by a Mittag–Leffler process induced by an associated Harris chain, at the discrete-time level. Previous results in Owada and Samorodnitsky [Ann. Probab. 43 (2015) 240–285] dealt with positive dependence in the increment process, whereas this paper derives the functional limit theorems under negative dependence. The negative dependence is due to cancellations arising from Gaussian-type fluctuations of functionals of the associated Harris chain. The new types of limiting processes involve stable random measures, due to heavy tails, Mittag–Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations. Along the way, we prove a function central limit theorem for fluctuations of functionals of Harris chains which is of independent interest as it extends a result of Chen [Probab. Theory Related Fields 116 (2000) 89–123].","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"2087-2130"},"PeriodicalIF":2.3,"publicationDate":"2017-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1107","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46921277","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}
We establish a Krylov-type estimate and an Ito–Krylov change of variable formula for the solutions of one-dimensional quadratic backward stochastic differential equations (QBSDEs) with a measurable generator and an arbitrary terminal datum. This allows us to prove various existence and uniqueness results for some classes of QBSDEs with a square integrable terminal condition and sometimes a merely measurable generator. It turns out that neither the existence of exponential moments of the terminal datum nor the continuity of the generator are necessary to the existence and/or uniqueness of solutions. We also establish a comparison theorem for solutions of a particular class of QBSDEs with measurable generator. As a byproduct, we obtain the existence of viscosity solutions for a particular class of quadratic partial differential equations (QPDEs) with a square integrable terminal datum.
{"title":"Quadratic BSDE with $mathbb{L}^{2}$-terminal data: Krylov’s estimate, Itô–Krylov’s formula and existence results","authors":"K. Bahlali, M. Eddahbi, Y. Ouknine","doi":"10.1214/16-AOP1115","DOIUrl":"https://doi.org/10.1214/16-AOP1115","url":null,"abstract":"We establish a Krylov-type estimate and an Ito–Krylov change of variable formula for the solutions of one-dimensional quadratic backward stochastic differential equations (QBSDEs) with a measurable generator and an arbitrary terminal datum. This allows us to prove various existence and uniqueness results for some classes of QBSDEs with a square integrable terminal condition and sometimes a merely measurable generator. It turns out that neither the existence of exponential moments of the terminal datum nor the continuity of the generator are necessary to the existence and/or uniqueness of solutions. We also establish a comparison theorem for solutions of a particular class of QBSDEs with measurable generator. As a byproduct, we obtain the existence of viscosity solutions for a particular class of quadratic partial differential equations (QPDEs) with a square integrable terminal datum.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"2377-2397"},"PeriodicalIF":2.3,"publicationDate":"2017-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1115","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46997673","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}
Consider the following nonlocal integro-differential operator: for α∈(0,2)α∈(0,2):L(α)σ,bf(x):=p.v.∫|z|<δf(x+σ(x)z)−f(x)|z|d+αdz+b(x)⋅∇f(x)+Lf(x),Lσ,b(α)f(x):=p.v.∫|z|<δf(x+σ(x)z)−f(x)|z|d+αdz+b(x)⋅∇f(x)+Lf(x),where σ:Rd→Rd⊗Rdσ:Rd→Rd⊗Rd and b:Rd→Rdb:Rd→Rd are smooth functions and have bounded partial derivatives of all orders greater than 11, δδ is a small positive number, p.v. stands for the Cauchy principal value and LL is a bounded linear operator in Sobolev spaces. Let B1(x):=σ(x)B1(x):=σ(x) and Bj+1(x):=b(x)⋅∇Bj(x)−∇b(x)⋅Bj(x)Bj+1(x):=b(x)⋅∇Bj(x)−∇b(x)⋅Bj(x) for j∈Nj∈N. Suppose Bj∈C∞b(Rd;Rd⊗Rd)Bj∈Cb∞(Rd;Rd⊗Rd) for each j∈Nj∈N. Under the following uniform Hormander’s type condition: for some j0∈Nj0∈N,infx∈Rdinf|u|=1∑j=1j0|uBj(x)|2>0,infx∈Rdinf|u|=1∑j=1j0|uBj(x)|2>0,by using Bismut’s approach to the Malliavin calculus with jumps, we prove the existence of fundamental solutions to operator L(α)σ,bLσ,b(α). In particular, we answer a question proposed by Nualart [Sankhyā A 73 (2011) 46–49] and Varadhan [Sankhyā A 73 (2011) 50–51].
{"title":"Fundamental solutions of nonlocal Hörmander’s operators II","authors":"Xicheng Zhang","doi":"10.1214/16-AOP1102","DOIUrl":"https://doi.org/10.1214/16-AOP1102","url":null,"abstract":"Consider the following nonlocal integro-differential operator: for α∈(0,2)α∈(0,2):L(α)σ,bf(x):=p.v.∫|z|<δf(x+σ(x)z)−f(x)|z|d+αdz+b(x)⋅∇f(x)+Lf(x),Lσ,b(α)f(x):=p.v.∫|z|<δf(x+σ(x)z)−f(x)|z|d+αdz+b(x)⋅∇f(x)+Lf(x),where σ:Rd→Rd⊗Rdσ:Rd→Rd⊗Rd and b:Rd→Rdb:Rd→Rd are smooth functions and have bounded partial derivatives of all orders greater than 11, δδ is a small positive number, p.v. stands for the Cauchy principal value and LL is a bounded linear operator in Sobolev spaces. Let B1(x):=σ(x)B1(x):=σ(x) and Bj+1(x):=b(x)⋅∇Bj(x)−∇b(x)⋅Bj(x)Bj+1(x):=b(x)⋅∇Bj(x)−∇b(x)⋅Bj(x) for j∈Nj∈N. Suppose Bj∈C∞b(Rd;Rd⊗Rd)Bj∈Cb∞(Rd;Rd⊗Rd) for each j∈Nj∈N. Under the following uniform Hormander’s type condition: for some j0∈Nj0∈N,infx∈Rdinf|u|=1∑j=1j0|uBj(x)|2>0,infx∈Rdinf|u|=1∑j=1j0|uBj(x)|2>0,by using Bismut’s approach to the Malliavin calculus with jumps, we prove the existence of fundamental solutions to operator L(α)σ,bLσ,b(α). In particular, we answer a question proposed by Nualart [Sankhyā A 73 (2011) 46–49] and Varadhan [Sankhyā A 73 (2011) 50–51].","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"1799-1841"},"PeriodicalIF":2.3,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1102","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42248381","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}
We prove that, for (adapted) stationary processes, the so-called Maxwell–Woodroofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. That result actually holds in the context of Banach valued stationary processes, including the case of $L^{p}$-valued random variables, with $1le p
我们证明，对于(适应的)平稳过程，所谓的Maxwell-Woodroofe条件对于迭代对数律是充分的，并且在某种意义上是最优的。该结果实际上适用于巴拿赫值平稳过程，包括$L^{p}$值随机变量的情况，$1le p
{"title":"Invariance principles under the Maxwell–Woodroofe condition in Banach spaces","authors":"C. Cuny","doi":"10.1214/16-AOP1095","DOIUrl":"https://doi.org/10.1214/16-AOP1095","url":null,"abstract":"We prove that, for (adapted) stationary processes, the so-called Maxwell–Woodroofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. That result actually holds in the context of Banach valued stationary processes, including the case of $L^{p}$-valued random variables, with $1le p<infty$. In this setting, we also prove the weak invariance principle, hence generalizing a result of Peligrad and Utev [Ann. Probab. 33 (2005) 798–815]. The proofs make use of a new maximal inequality and of approximation by martingales, for which some of our results are also new.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"1578-1611"},"PeriodicalIF":2.3,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1095","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48762318","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}
We prove a structural result for degree-dd polynomials. In particular, we show that any degree-dd polynomial, pp can be approximated by another polynomial, p0p0, which can be decomposed as some function of polynomials q1,…,qmq1,…,qm with qiqi normalized and m=Od(1)m=Od(1), so that if XX is a Gaussian random variable, the probability distribution on (q1(X),…,qm(X))(q1(X),…,qm(X)) does not have too much mass in any small box. Using this result, we prove improved versions of a number of results about polynomial threshold functions, including producing better pseudorandom generators, obtaining a better invariance principle, and proving improved bounds on noise sensitivity.
{"title":"A structure theorem for poorly anticoncentrated polynomials of Gaussians and applications to the study of polynomial threshold functions","authors":"D. Kane","doi":"10.1214/16-AOP1097","DOIUrl":"https://doi.org/10.1214/16-AOP1097","url":null,"abstract":"We prove a structural result for degree-dd polynomials. In particular, we show that any degree-dd polynomial, pp can be approximated by another polynomial, p0p0, which can be decomposed as some function of polynomials q1,…,qmq1,…,qm with qiqi normalized and m=Od(1)m=Od(1), so that if XX is a Gaussian random variable, the probability distribution on (q1(X),…,qm(X))(q1(X),…,qm(X)) does not have too much mass in any small box. \u0000 \u0000Using this result, we prove improved versions of a number of results about polynomial threshold functions, including producing better pseudorandom generators, obtaining a better invariance principle, and proving improved bounds on noise sensitivity.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"1612-1679"},"PeriodicalIF":2.3,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1097","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46250420","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}
Etant donn{e}e une filtration $(zc_n)_{n le 0}$ index{e}e par les entiers n{e}gatifs, nous introduisons la notion de compl{e}mentabilit{e} pour les filtrations incluses dans $(zc_n)_{n le 0}$. Dans le cas de filtrations poly-adiques, nous d{e}finissons aussi la notion de maximalit{e}, dont nous donnons plusieurs caract{e}risations. Lorsque $(zc_n)_{n le 0}$ est poly-adique, nous montrons que toute filtration compl{e}mentable par une filtration kolmogorovienne est maximale dans $(zc_n)_{n le 0}$. Nous montrons que la r{e}ciproque est fausse, mais qu'une r{e}ciproque partielle est vraie. Cette r{e}ciproque partielle {e}tend le th{e}or{e}me d'isomorphisme lacunaire de Vershik dans le cas des filtrations poly-adiques.
{"title":"Poly-adic filtrations, standardness, complementability and maximality","authors":"Christophe Leuridan","doi":"10.1214/15-AOP1085","DOIUrl":"https://doi.org/10.1214/15-AOP1085","url":null,"abstract":"Etant donn{e}e une filtration $(zc_n)_{n le 0}$ index{e}e par les entiers n{e}gatifs, nous introduisons la notion de compl{e}mentabilit{e} pour les filtrations incluses dans $(zc_n)_{n le 0}$. Dans le cas de filtrations poly-adiques, nous d{e}finissons aussi la notion de maximalit{e}, dont nous donnons plusieurs caract{e}risations. Lorsque $(zc_n)_{n le 0}$ est poly-adique, nous montrons que toute filtration compl{e}mentable par une filtration kolmogorovienne est maximale dans $(zc_n)_{n le 0}$. Nous montrons que la r{e}ciproque est fausse, mais qu'une r{e}ciproque partielle est vraie. Cette r{e}ciproque partielle {e}tend le th{e}or{e}me d'isomorphisme lacunaire de Vershik dans le cas des filtrations poly-adiques.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"1218-1246"},"PeriodicalIF":2.3,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1085","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44029705","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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