We study random surfaces with a uniformly convex gradient interaction in the presence of quenched disorder taking the form of a random independent external field. Previous work on the model has focused on proving existence and uniqueness of infinite-volume gradient Gibbs measures with a given tilt and on studying the fluctuations of the surface and its discrete gradient. In this work we focus on the convergence of the thermodynamic limit, establishing convergence of the finite-volume distributions with Dirichlet boundary conditions to translation-covariant (gradient) Gibbs measures. Specifically, it is shown that, when the law of the random field has finite second moment and is symmetric, the distribution of the gradient of the surface converges in dimensions $dgeq4$ while the distribution of the surface itself converges in dimensions $dgeq 5$. Moreover, a power-law upper bound on the rate of convergence in Wasserstein distance is obtained. The results partially answer a question discussed by Cotar and K"ulske
{"title":"Convergence to the thermodynamic limit for random-field random surfaces","authors":"P. Dario","doi":"10.1214/22-aap1844","DOIUrl":"https://doi.org/10.1214/22-aap1844","url":null,"abstract":"We study random surfaces with a uniformly convex gradient interaction in the presence of quenched disorder taking the form of a random independent external field. Previous work on the model has focused on proving existence and uniqueness of infinite-volume gradient Gibbs measures with a given tilt and on studying the fluctuations of the surface and its discrete gradient. In this work we focus on the convergence of the thermodynamic limit, establishing convergence of the finite-volume distributions with Dirichlet boundary conditions to translation-covariant (gradient) Gibbs measures. Specifically, it is shown that, when the law of the random field has finite second moment and is symmetric, the distribution of the gradient of the surface converges in dimensions $dgeq4$ while the distribution of the surface itself converges in dimensions $dgeq 5$. Moreover, a power-law upper bound on the rate of convergence in Wasserstein distance is obtained. The results partially answer a question discussed by Cotar and K\"ulske","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41644161","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}
Christian Bayer, Paul Hager, Sebastian Riedel, J. Schoenmakers
We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process $X$. We consider classic and randomized stopping times represented by linear and non-linear functionals of the rough path signature $mathbb{X}^{
{"title":"Optimal stopping with signatures","authors":"Christian Bayer, Paul Hager, Sebastian Riedel, J. Schoenmakers","doi":"10.1214/22-aap1814","DOIUrl":"https://doi.org/10.1214/22-aap1814","url":null,"abstract":"We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process $X$. We consider classic and randomized stopping times represented by linear and non-linear functionals of the rough path signature $mathbb{X}^{<infty}$ associated to $X$, and prove that maximizing over these classes of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature $mathbb{E}left[ mathbb{X}^{le N}_{0,T} right]$. By applying a deep neural network approach to approximate the non-linear signature functionals, we can efficiently solve the optimal stopping problem numerically. The only assumption on the process $X$ is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion, which fail to be either semi-martingales or Markov processes, and can be used, in particular, for American-type option pricing in fractional models, e.g. on financial or electricity markets.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43101412","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}
Time-evolving random graph models have appeared and have been studied in various fields of research over the past decades. However, the rigorous mathematical treatment of large graphs and their limits at the process-level is still in its infancy. In this article, we adapt the approach of Athreya, den Hollander and R"ollin (2021+) to the setting of multigraphs and multigraphons, introduced by Kolossv'ary and R'ath (2011). We then generalise the work of R'ath (2012) and R'ath and Szak'acs (2012), who analysed edge-flipping dynamics on the configuration model -- in contrast to their work, we establish weak convergence at the process-level, and by allowing removal and addition of edges, these limits are non-deterministic.
{"title":"Dense multigraphon-valued stochastic processes and edge-changing dynamics in the configuration model","authors":"A. Rollin, Zhuohui Zhang","doi":"10.1214/22-aap1889","DOIUrl":"https://doi.org/10.1214/22-aap1889","url":null,"abstract":"Time-evolving random graph models have appeared and have been studied in various fields of research over the past decades. However, the rigorous mathematical treatment of large graphs and their limits at the process-level is still in its infancy. In this article, we adapt the approach of Athreya, den Hollander and R\"ollin (2021+) to the setting of multigraphs and multigraphons, introduced by Kolossv'ary and R'ath (2011). We then generalise the work of R'ath (2012) and R'ath and Szak'acs (2012), who analysed edge-flipping dynamics on the configuration model -- in contrast to their work, we establish weak convergence at the process-level, and by allowing removal and addition of edges, these limits are non-deterministic.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43748730","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 mean-field Zero-Range process in the regime where the potential function $r$ is increasing to infinity at sublinear speed, and the density of particles is bounded. We determine the mixing time of the system, and establish cutoff. We also prove that the Poincare constant is bounded away from zero and infinity. This mean-field estimate extends to arbitrary geometries via a comparison argument. Our proof uses the path-coupling method of Bubley and Dyer and stochastic calculus.
{"title":"The mean-field zero-range process with unbounded monotone rates: Mixing time, cutoff, and Poincaré constant","authors":"Hong-Quan Tran","doi":"10.1214/22-aap1851","DOIUrl":"https://doi.org/10.1214/22-aap1851","url":null,"abstract":"We consider the mean-field Zero-Range process in the regime where the potential function $r$ is increasing to infinity at sublinear speed, and the density of particles is bounded. We determine the mixing time of the system, and establish cutoff. We also prove that the Poincare constant is bounded away from zero and infinity. This mean-field estimate extends to arbitrary geometries via a comparison argument. Our proof uses the path-coupling method of Bubley and Dyer and stochastic calculus.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48705028","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}
If a given aggregate process S is a mixed compound Poisson process under a probability measure P , we provide a characterization of all probability measures Q on the domain of P , such that P and Q are progressively equivalent and S remains a mixed compound Poisson process with improved properties. This result generalizes earlier work of Delbaen & Haezendonck (1989). Implications related to the computation of premium calculation principles in an insurance market possessing the property of no free lunch with vanishing risk are also discussed.
{"title":"A characterization of martingale-equivalent mixed compound Poisson processes","authors":"D. P. Lyberopoulos, N. D. Macheras","doi":"10.1214/20-AAP1604","DOIUrl":"https://doi.org/10.1214/20-AAP1604","url":null,"abstract":"If a given aggregate process S is a mixed compound Poisson process under a probability measure P , we provide a characterization of all probability measures Q on the domain of P , such that P and Q are progressively equivalent and S remains a mixed compound Poisson process with improved properties. This result generalizes earlier work of Delbaen & Haezendonck (1989). Implications related to the computation of premium calculation principles in an insurance market possessing the property of no free lunch with vanishing risk are also discussed.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85819654","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 dynamic large deviation behaviour of Kac's collisional process for a range of initial conditions including equilibrium. We prove an upper bound with a rate function of the type which has previously been found for kinetic large deviation problems, and a matching lower bound restricted to a class of sufficiently good paths. However, we are able to show by an explicit counterexample that the predicted rate function does not extend to a global lower bound: even though the particle system almost surely conserves energy, large deviation behaviour includes solutions to the Boltzmann equation which do not conserve energy, as found by Lu and Wennberg, and these occur strictly more rarely than predicted by the proposed rate function. At the level of the particle system, this occurs because a macroscopic proportion of energy can concentrate in $mathfrak{o}(N)$ particles with probability $e^{-mathcal{O}(N)}$.
{"title":"Large deviations of Kac’s conservative particle system and energy nonconserving solutions to the Boltzmann equation: A counterexample to the predicted rate function","authors":"Daniel Heydecker","doi":"10.1214/22-aap1852","DOIUrl":"https://doi.org/10.1214/22-aap1852","url":null,"abstract":"We consider the dynamic large deviation behaviour of Kac's collisional process for a range of initial conditions including equilibrium. We prove an upper bound with a rate function of the type which has previously been found for kinetic large deviation problems, and a matching lower bound restricted to a class of sufficiently good paths. However, we are able to show by an explicit counterexample that the predicted rate function does not extend to a global lower bound: even though the particle system almost surely conserves energy, large deviation behaviour includes solutions to the Boltzmann equation which do not conserve energy, as found by Lu and Wennberg, and these occur strictly more rarely than predicted by the proposed rate function. At the level of the particle system, this occurs because a macroscopic proportion of energy can concentrate in $mathfrak{o}(N)$ particles with probability $e^{-mathcal{O}(N)}$.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47468861","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 objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Čech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider Čech filtration over a scaled random sample r−1 n Xn = {r−1 n X1, . . . , r−1 n Xn}, such that rn → 0 as n → ∞. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: nr n → 0, n → ∞. In this setting, we show that the asymptotics of the kth persistence diagram depends on the limit value of the sequence nr d(k+1) n . If n r d(k+1) n → ∞, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If rn decays faster so that nr d(k+1) n → c ∈ (0,∞), the persistence diagram weakly converges to a limiting point process without normalization. Finally, if nr d(k+1) n → 0, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the M0-topology.
本文的目的是研究与Čech过滤相关的持续图的渐近行为。持久性图是几何对象的拓扑和代数结构的图形描述符。我们考虑Čech过滤在一个缩放的随机样本r−1 n Xn = {r−1 n X1，…， r−1 n Xn}，使得rn→0 = n→∞。我们将持久性图视为一个点过程，并建立了其在稀疏域的极限定理:nr n→0,n→∞。在这种情况下，我们证明了第k个持续图的渐近性取决于序列nr d(k+1) n的极限值。当n r d(k+1) n→∞时，尺度持续图在模糊度量中几乎肯定收敛于确定性Radon测度。如果rn衰减较快，使得nr d(k+1) n→c∈(0，∞)，则持久性图弱收敛到一个不归一化的极限点过程。最后，如果nr d(k+1) n→0，则持久性图的概率分布序列应归一化，并根据m0拓扑处理由此产生的收敛性。
{"title":"Convergence of persistence diagram in the sparse regime","authors":"Takashi Owada","doi":"10.1214/22-aap1800","DOIUrl":"https://doi.org/10.1214/22-aap1800","url":null,"abstract":"The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Čech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider Čech filtration over a scaled random sample r−1 n Xn = {r−1 n X1, . . . , r−1 n Xn}, such that rn → 0 as n → ∞. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: nr n → 0, n → ∞. In this setting, we show that the asymptotics of the kth persistence diagram depends on the limit value of the sequence nr d(k+1) n . If n r d(k+1) n → ∞, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If rn decays faster so that nr d(k+1) n → c ∈ (0,∞), the persistence diagram weakly converges to a limiting point process without normalization. Finally, if nr d(k+1) n → 0, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the M0-topology.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48906408","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 get some convergence rates in total variation distance in approximating discretized paths of L{'e}vy driven stochastic differential equations, assuming that the driving process is locally stable. The particular case of the Euler approximation is studied. Our results are based on sharp local estimates in Hellinger distance obtained using Malliavin calculus for jump processes.
{"title":"Hellinger and total variation distance in approximating Lévy driven SDEs","authors":"E. Cl'ement","doi":"10.1214/22-aap1863","DOIUrl":"https://doi.org/10.1214/22-aap1863","url":null,"abstract":"In this paper, we get some convergence rates in total variation distance in approximating discretized paths of L{'e}vy driven stochastic differential equations, assuming that the driving process is locally stable. The particular case of the Euler approximation is studied. Our results are based on sharp local estimates in Hellinger distance obtained using Malliavin calculus for jump processes.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43552147","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 propose and study a new model for competitions, specifically sports multi-player leagues where the initial strengths of the teams are independent i.i.d. random variables that evolve during different days of the league according to independent ergodic processes. The result of each match is random: the probability that a team wins against another team is determined by a function of the strengths of the two teams in the day the match is played. Our model generalizes some previous models studied in the physical and mathematical literature and is defined in terms of different parameters that can be statistically calibrated. We prove a quenched -- conditioning on the initial strengths of the teams -- law of large numbers and a quenched central limit theorem for the number of victories of a team according to its initial strength. To obtain our results, we prove a theorem of independent interest. For a stationary process $xi=(xi_i)_{iin mathbb{N}}$ satisfying a mixing condition and an independent sequence of i.i.d. random variables $(s_i)_{iin mathbb{N}}$, we prove a quenched -- conditioning on $(s_i)_{iinmathbb{N}}$ -- central limit theorem for sums of the form $sum_{i=1}^{n}gleft(xi_i,s_iright)$, where $g$ is a bounded measurable function. We highlight that the random variables $gleft(xi_i,s_iright)$ are not stationary conditioning on $(s_i)_{iinmathbb{N}}$.
{"title":"Quenched law of large numbers and quenched central limit theorem for multiplayer leagues with ergodic strengths","authors":"J. Borga, Benedetta Cavalli","doi":"10.1214/22-aap1790","DOIUrl":"https://doi.org/10.1214/22-aap1790","url":null,"abstract":"We propose and study a new model for competitions, specifically sports multi-player leagues where the initial strengths of the teams are independent i.i.d. random variables that evolve during different days of the league according to independent ergodic processes. The result of each match is random: the probability that a team wins against another team is determined by a function of the strengths of the two teams in the day the match is played. Our model generalizes some previous models studied in the physical and mathematical literature and is defined in terms of different parameters that can be statistically calibrated. We prove a quenched -- conditioning on the initial strengths of the teams -- law of large numbers and a quenched central limit theorem for the number of victories of a team according to its initial strength. To obtain our results, we prove a theorem of independent interest. For a stationary process $xi=(xi_i)_{iin mathbb{N}}$ satisfying a mixing condition and an independent sequence of i.i.d. random variables $(s_i)_{iin mathbb{N}}$, we prove a quenched -- conditioning on $(s_i)_{iinmathbb{N}}$ -- central limit theorem for sums of the form $sum_{i=1}^{n}gleft(xi_i,s_iright)$, where $g$ is a bounded measurable function. We highlight that the random variables $gleft(xi_i,s_iright)$ are not stationary conditioning on $(s_i)_{iinmathbb{N}}$.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46506912","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 employ stabilization methods and second order Poincar'e inequalities to establish rates of multivariate normal convergence for a large class of vectors $(H_s^{(1)},...,H_s^{(m)})$, $s geq 1$, of statistics of marked Poisson processes on $mathbb{R}^d$, $d geq 2$, as the intensity parameter $s$ tends to infinity. Our results are applicable whenever the constituent functionals $H_s^{(i)}$, $iin{1,...,m}$, are expressible as sums of exponentially stabilizing score functions satisfying a moment condition. The rates are for the $d_2$-, $d_3$-, and $d_{convex}$-distances. When we compare with a centered Gaussian random vector, whose covariance matrix is given by the asymptotic covariances, the rates are in general unimprovable and are governed by the rate of convergence of $s^{-1} {rm Cov}( H_s^{(i)}, H_s^{(j)})$, $i,jin{1,...,m}$, to the limiting covariance, shown to be of order $s^{-1/d}$. We use the general results to deduce rates of multivariate normal convergence for statistics arising in random graphs and topological data analysis as well as for multivariate statistics used to test equality of distributions. Some of our results hold for stabilizing functionals of Poisson input on suitable metric spaces.
{"title":"Rates of multivariate normal approximation for statistics in geometric probability","authors":"Matthias Schulte, J. Yukich","doi":"10.1214/22-aap1822","DOIUrl":"https://doi.org/10.1214/22-aap1822","url":null,"abstract":"We employ stabilization methods and second order Poincar'e inequalities to establish rates of multivariate normal convergence for a large class of vectors $(H_s^{(1)},...,H_s^{(m)})$, $s geq 1$, of statistics of marked Poisson processes on $mathbb{R}^d$, $d geq 2$, as the intensity parameter $s$ tends to infinity. Our results are applicable whenever the constituent functionals $H_s^{(i)}$, $iin{1,...,m}$, are expressible as sums of exponentially stabilizing score functions satisfying a moment condition. The rates are for the $d_2$-, $d_3$-, and $d_{convex}$-distances. When we compare with a centered Gaussian random vector, whose covariance matrix is given by the asymptotic covariances, the rates are in general unimprovable and are governed by the rate of convergence of $s^{-1} {rm Cov}( H_s^{(i)}, H_s^{(j)})$, $i,jin{1,...,m}$, to the limiting covariance, shown to be of order $s^{-1/d}$. We use the general results to deduce rates of multivariate normal convergence for statistics arising in random graphs and topological data analysis as well as for multivariate statistics used to test equality of distributions. Some of our results hold for stabilizing functionals of Poisson input on suitable metric spaces.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45102554","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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