We study the discrepancy between the distribution of a vector-valued functional of i.i.d. random elements and that of a Gaussian vector. Our main contribution is an explicit bound on the convex distance between the two distributions, holding in every dimension. Such a finding constitutes a substantial extension of the one-dimensional bounds deduced in Chatterjee (Ann. Probab. 36 (2008) 1584–1610) and Lachièze-Rey and Peccati (Ann. Appl. Probab. 27 (2017) 1992–2031), as well as of the multidimensional bounds for smooth test functions and indicators of rectangles derived, respectively, in Dung (Acta Math. Hungar. 158 (2019) 173–201), and Fang and Koike (Ann. Appl. Probab. 31 (2021) 1660–1686). Our techniques involve the use of Stein’s method, combined with a suitable adaptation of the recursive approach inaugurated by Schulte and Yukich (Electron. J. Probab. 24 (2019) 1–42): this yields rates of converge that have a presumably optimal dependence on the sample size. We develop several applications of a geometric nature, among which is a new collection of multidimensional quantitative limit theorems for the intrinsic volumes associated with coverage processes in Euclidean spaces.
{"title":"Vector-valued statistics of binomial processes: Berry–Esseen bounds in the convex distance","authors":"Mikołaj J. Kasprzak, Giovanni Peccati","doi":"10.1214/22-aap1897","DOIUrl":"https://doi.org/10.1214/22-aap1897","url":null,"abstract":"We study the discrepancy between the distribution of a vector-valued functional of i.i.d. random elements and that of a Gaussian vector. Our main contribution is an explicit bound on the convex distance between the two distributions, holding in every dimension. Such a finding constitutes a substantial extension of the one-dimensional bounds deduced in Chatterjee (Ann. Probab. 36 (2008) 1584–1610) and Lachièze-Rey and Peccati (Ann. Appl. Probab. 27 (2017) 1992–2031), as well as of the multidimensional bounds for smooth test functions and indicators of rectangles derived, respectively, in Dung (Acta Math. Hungar. 158 (2019) 173–201), and Fang and Koike (Ann. Appl. Probab. 31 (2021) 1660–1686). Our techniques involve the use of Stein’s method, combined with a suitable adaptation of the recursive approach inaugurated by Schulte and Yukich (Electron. J. Probab. 24 (2019) 1–42): this yields rates of converge that have a presumably optimal dependence on the sample size. We develop several applications of a geometric nature, among which is a new collection of multidimensional quantitative limit theorems for the intrinsic volumes associated with coverage processes in Euclidean spaces.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136168582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We first establish new local limit estimates for the probability that a nondecreasing integer-valued random walk lies at time n at an arbitrary value, encompassing in particular large deviation regimes on the boundary of the Cramér zone. This enables us to derive scaling limits of such random walks conditioned by their terminal value at time n in various regimes. We believe both to be of independent interest. We then apply these results to obtain invariance principles for the Łukasiewicz path of Bienaymé–Galton–Watson trees conditioned on having a fixed number of leaves and of vertices at the same time, which constitutes a first step towards understanding their large scale geometry. We finally deduce from this scaling limit theorems for random bipartite planar maps under a new conditioning by fixing their number of vertices, edges, and faces at the same time. In the particular case of the uniform distribution, our results confirm a prediction of Fusy and Guitter on the growth of the typical distances and show furthermore that in all regimes, the scaling limit is the celebrated Brownian sphere.
{"title":"Large deviation local limit theorems and limits of biconditioned planar maps","authors":"Igor Kortchemski, Cyril Marzouk","doi":"10.1214/22-aap1906","DOIUrl":"https://doi.org/10.1214/22-aap1906","url":null,"abstract":"We first establish new local limit estimates for the probability that a nondecreasing integer-valued random walk lies at time n at an arbitrary value, encompassing in particular large deviation regimes on the boundary of the Cramér zone. This enables us to derive scaling limits of such random walks conditioned by their terminal value at time n in various regimes. We believe both to be of independent interest. We then apply these results to obtain invariance principles for the Łukasiewicz path of Bienaymé–Galton–Watson trees conditioned on having a fixed number of leaves and of vertices at the same time, which constitutes a first step towards understanding their large scale geometry. We finally deduce from this scaling limit theorems for random bipartite planar maps under a new conditioning by fixing their number of vertices, edges, and faces at the same time. In the particular case of the uniform distribution, our results confirm a prediction of Fusy and Guitter on the growth of the typical distances and show furthermore that in all regimes, the scaling limit is the celebrated Brownian sphere.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the problem of computing with many coins of unknown bias. We are given access to samples of n coins with unknown biases p1,…,pn and are asked to sample from a coin with bias f(p1,…,pn) for a given function f:[0,1]n→[0,1]. We give a complete characterization of the functions f for which this is possible. As a consequence, we show how to extend various combinatorial sampling procedures (most notably, the classic Sampford sampling for k-subsets) to the boundary of the hypercube.
{"title":"Multiparameter Bernoulli factories","authors":"Renato Paes Leme, Jon Schneider","doi":"10.1214/22-aap1913","DOIUrl":"https://doi.org/10.1214/22-aap1913","url":null,"abstract":"We consider the problem of computing with many coins of unknown bias. We are given access to samples of n coins with unknown biases p1,…,pn and are asked to sample from a coin with bias f(p1,…,pn) for a given function f:[0,1]n→[0,1]. We give a complete characterization of the functions f for which this is possible. As a consequence, we show how to extend various combinatorial sampling procedures (most notably, the classic Sampford sampling for k-subsets) to the boundary of the hypercube.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136167555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the stochastic nonlinear Schrödinger equations with linear multiplicative noise, particularly in the defocusing mass-critical and energy-critical cases. For general initial data, we prove the global well-posedness of solutions in both mass-critical and energy-critical cases. We also prove the rescaled scattering behavior of global solutions in the spaces L2, H1 as well as the pseudo-conformal space for dimensions d≥3 in the case of finite global quadratic variation of noise. Furthermore, the Stroock–Varadhan type theorem is also obtained for the topological support of the probability distribution induced by global solutions in the Strichartz and local smoothing spaces. Our proof is based on the construction of a new family of rescaling transformations indexed by stopping times and on the stability analysis adapted to the multiplicative noise.
{"title":"Stochastic nonlinear Schrödinger equations in the defocusing mass and energy critical cases","authors":"Deng Zhang","doi":"10.1214/22-aap1903","DOIUrl":"https://doi.org/10.1214/22-aap1903","url":null,"abstract":"We study the stochastic nonlinear Schrödinger equations with linear multiplicative noise, particularly in the defocusing mass-critical and energy-critical cases. For general initial data, we prove the global well-posedness of solutions in both mass-critical and energy-critical cases. We also prove the rescaled scattering behavior of global solutions in the spaces L2, H1 as well as the pseudo-conformal space for dimensions d≥3 in the case of finite global quadratic variation of noise. Furthermore, the Stroock–Varadhan type theorem is also obtained for the topological support of the probability distribution induced by global solutions in the Strichartz and local smoothing spaces. Our proof is based on the construction of a new family of rescaling transformations indexed by stopping times and on the stability analysis adapted to the multiplicative noise.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136168196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yaozhong Hu, Michael A. Kouritzin, Panqiu Xia, Jiayu Zheng
The mean-field stochastic partial differential equation (SPDE) corresponding to a mean-field super-Brownian motion (sBm) is obtained and studied. In this mean-field sBm, the branching-particle lifetime is allowed to depend upon the probability distribution of the sBm itself, producing an SPDE whose space-time white noise coefficient has, in addition to the typical sBm square root, an extra factor that is a function of the probability law of the density of the mean-field sBm. This novel mean-field SPDE is thus motivated by population models where things like overcrowding and isolation can affect growth. A two step approximation method is employed to show the existence for this SPDE under general conditions. Then, mild moment conditions are imposed to get uniqueness. Finally, smoothness of the SPDE solution is established under a further simplifying condition.
{"title":"On mean-field super-Brownian motions","authors":"Yaozhong Hu, Michael A. Kouritzin, Panqiu Xia, Jiayu Zheng","doi":"10.1214/22-aap1909","DOIUrl":"https://doi.org/10.1214/22-aap1909","url":null,"abstract":"The mean-field stochastic partial differential equation (SPDE) corresponding to a mean-field super-Brownian motion (sBm) is obtained and studied. In this mean-field sBm, the branching-particle lifetime is allowed to depend upon the probability distribution of the sBm itself, producing an SPDE whose space-time white noise coefficient has, in addition to the typical sBm square root, an extra factor that is a function of the probability law of the density of the mean-field sBm. This novel mean-field SPDE is thus motivated by population models where things like overcrowding and isolation can affect growth. A two step approximation method is employed to show the existence for this SPDE under general conditions. Then, mild moment conditions are imposed to get uniqueness. Finally, smoothness of the SPDE solution is established under a further simplifying condition.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136167556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 study expected utility maximization under ratchet and drawdown constraints on consumption in a general incomplete semimartingale market using duality methods. The optimization is considered with respect to two parameters: the initial wealth and the essential lower bound on consumption process. In order to state the problem and define the primal domains, we introduce a natural extension of the notion of running maximum to arbitrary nonnegative optional processes and study its properties. The dual domains for optimization are characterized in terms of solidity with respect to an ordering that is introduced on the set of nonnegative optional processes. The abstract duality result we obtain for the optimization problem is used in order to derive a more detailed characterization of solutions in the complete market case.
{"title":"Utility maximization with ratchet and drawdown constraints on consumption in incomplete semimartingale markets","authors":"Anastasiya Tanana","doi":"10.1214/22-aap1918","DOIUrl":"https://doi.org/10.1214/22-aap1918","url":null,"abstract":"In this paper, we study expected utility maximization under ratchet and drawdown constraints on consumption in a general incomplete semimartingale market using duality methods. The optimization is considered with respect to two parameters: the initial wealth and the essential lower bound on consumption process. In order to state the problem and define the primal domains, we introduce a natural extension of the notion of running maximum to arbitrary nonnegative optional processes and study its properties. The dual domains for optimization are characterized in terms of solidity with respect to an ordering that is introduced on the set of nonnegative optional processes. The abstract duality result we obtain for the optimization problem is used in order to derive a more detailed characterization of solutions in the complete market case.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The family of SLE4⟨μ⟩(−2) exploration processes with parameter μ∈R forms a natural class of conformally invariant ways for discovering the loops of a conformal loop ensemble CLE4. Such an exploration consists of one simple continuous path called the trunk of the exploration that discovers CLE4 loops along the way. The parameter μ appears in the Loewner chain description of the path that traces the trunk and all CLE4 loops encountered by the trunk in chronological order. These explorations can also be interpreted in terms of level lines of a Gaussian free field. It has been shown by Miller, Sheffield and Werner that the trunk of such an exploration is an SLE4(ρ,−2−ρ) process for some (unknown) value of ρ∈(−2,0). The main result of the present paper is to establish the relation between μ and ρ, more specifically to show that μ=−πcot(πρ/2).
{"title":"The trunks of CLE(4) explorations","authors":"Matthis Lehmkuehler","doi":"10.1214/22-aap1895","DOIUrl":"https://doi.org/10.1214/22-aap1895","url":null,"abstract":"The family of SLE4⟨μ⟩(−2) exploration processes with parameter μ∈R forms a natural class of conformally invariant ways for discovering the loops of a conformal loop ensemble CLE4. Such an exploration consists of one simple continuous path called the trunk of the exploration that discovers CLE4 loops along the way. The parameter μ appears in the Loewner chain description of the path that traces the trunk and all CLE4 loops encountered by the trunk in chronological order. These explorations can also be interpreted in terms of level lines of a Gaussian free field. It has been shown by Miller, Sheffield and Werner that the trunk of such an exploration is an SLE4(ρ,−2−ρ) process for some (unknown) value of ρ∈(−2,0). The main result of the present paper is to establish the relation between μ and ρ, more specifically to show that μ=−πcot(πρ/2).","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a large deviation principle for the point process associated to k-element connected components in Rd with respect to the connectivity radii rn→∞. The random points are generated from a homogeneous Poisson point process or the corresponding binomial point process, so that (rn)n≥1 satisfies nkrnd(k−1)→∞ and nrnd→0 as n→∞ (i.e., sparse regime). The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function.
{"title":"Large deviation principle for geometric and topological functionals and associated point processes","authors":"Christian Hirsch, Takashi Owada","doi":"10.1214/22-aap1914","DOIUrl":"https://doi.org/10.1214/22-aap1914","url":null,"abstract":"We prove a large deviation principle for the point process associated to k-element connected components in Rd with respect to the connectivity radii rn→∞. The random points are generated from a homogeneous Poisson point process or the corresponding binomial point process, so that (rn)n≥1 satisfies nkrnd(k−1)→∞ and nrnd→0 as n→∞ (i.e., sparse regime). The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136167554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alexis Anagnostakis, Antoine Lejay, Denis Villemonais
We prove the convergence of the law of grid-valued random walks, which can be seen as time-space Markov chains, to the law of a general diffusion process. This includes processes with sticky features, reflecting or absorbing boundaries and skew behavior. We prove that the convergence occurs at any rate strictly inferior to (1/4)∧(1/p) in terms of the maximum cell size of the grid, for any p-Wasserstein distance. We also show that it is possible to achieve any rate strictly inferior to (1/2)∧(2/p) if the grid is adapted to the speed measure of the diffusion, which is optimal for p≤4. This result allows us to set up asymptotically optimal approximation schemes for general diffusion processes. Last, we experiment numerically on diffusions that exhibit various features.
{"title":"General diffusion processes as limit of time-space Markov chains","authors":"Alexis Anagnostakis, Antoine Lejay, Denis Villemonais","doi":"10.1214/22-aap1902","DOIUrl":"https://doi.org/10.1214/22-aap1902","url":null,"abstract":"We prove the convergence of the law of grid-valued random walks, which can be seen as time-space Markov chains, to the law of a general diffusion process. This includes processes with sticky features, reflecting or absorbing boundaries and skew behavior. We prove that the convergence occurs at any rate strictly inferior to (1/4)∧(1/p) in terms of the maximum cell size of the grid, for any p-Wasserstein distance. We also show that it is possible to achieve any rate strictly inferior to (1/2)∧(2/p) if the grid is adapted to the speed measure of the diffusion, which is optimal for p≤4. This result allows us to set up asymptotically optimal approximation schemes for general diffusion processes. Last, we experiment numerically on diffusions that exhibit various features.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a general approach to Stein’s method for approximating a random process in the path space D([0,T]→Rd) by a real continuous Gaussian process. We then use the approach in the context of processes that have a representation as integrals with respect to an underlying point process, deriving a general quantitative Gaussian approximation. The error bound is expressed in terms of couplings of the original process to processes generated from the reduced Palm measures associated with the point process. As applications, we study certain GI/GI/∞ queues in the “heavy traffic” regime.
{"title":"Stein’s method, Gaussian processes and Palm measures, with applications to queueing","authors":"A. D. Barbour, Nathan Ross, Guangqu Zheng","doi":"10.1214/22-aap1908","DOIUrl":"https://doi.org/10.1214/22-aap1908","url":null,"abstract":"We develop a general approach to Stein’s method for approximating a random process in the path space D([0,T]→Rd) by a real continuous Gaussian process. We then use the approach in the context of processes that have a representation as integrals with respect to an underlying point process, deriving a general quantitative Gaussian approximation. The error bound is expressed in terms of couplings of the original process to processes generated from the reduced Palm measures associated with the point process. As applications, we study certain GI/GI/∞ queues in the “heavy traffic” regime.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136167559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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