Consider first passage percolation with identical and independent weight distributions and first passage time ${rm T}$. In this paper, we study the upper tail large deviations $mathbb{P}({rm T}(0,nx)>n(mu+xi))$, for $xi>0$ and $xneq 0$ with a time constant $mu$ and a dimension $d$, for weights that satisfy a tail assumption $ beta_1exp{(-alpha t^r)}leq mathbb P(tau_e>t)leq beta_2exp{(-alpha t^r)}.$ When $rleq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $exp{(-(2dxi +o(1))n)}$. When $1n(mu+xi)$ is described by a localization of high weights around the origin. The picture changes for $rgeq d$ where the configuration is not anymore localized.
{"title":"A variational formula for large deviations in first-passage percolation under tail estimates","authors":"Clément Cosco, S. Nakajima","doi":"10.1214/22-aap1861","DOIUrl":"https://doi.org/10.1214/22-aap1861","url":null,"abstract":"Consider first passage percolation with identical and independent weight distributions and first passage time ${rm T}$. In this paper, we study the upper tail large deviations $mathbb{P}({rm T}(0,nx)>n(mu+xi))$, for $xi>0$ and $xneq 0$ with a time constant $mu$ and a dimension $d$, for weights that satisfy a tail assumption $ beta_1exp{(-alpha t^r)}leq mathbb P(tau_e>t)leq beta_2exp{(-alpha t^r)}.$ When $rleq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $exp{(-(2dxi +o(1))n)}$. When $1<rleq d$, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. For $r<d$, we show that the large deviation event ${rm T}(0,nx)>n(mu+xi)$ is described by a localization of high weights around the origin. The picture changes for $rgeq d$ where the configuration is not anymore localized.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41622328","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 novel procedure for estimating the optimizer of general convex stochastic optimization problems of the form minx∈X E[F (x, ξ)], when the given data is a finite independent sample selected according to ξ. The procedure is based on a median-of-means tournament, and is the first procedure that exhibits the optimal statistical performance in heavy tailed situations: we recover the asymptotic rates dictated by the central limit theorem in a non-asymptotic manner once the sample size exceeds some explicitly computable threshold. Additionally, our results apply in the high-dimensional setup, as the threshold sample size exhibits the optimal dependence on the dimension (up to a logarithmic factor). The general setting allows us to recover recent results on multivariate mean estimation and linear regression in heavy-tailed situations and to prove the first sharp, non-asymptotic results for the portfolio optimization problem.
{"title":"On Monte-Carlo methods in convex stochastic optimization","authors":"Daniel Bartl, S. Mendelson","doi":"10.1214/22-aap1781","DOIUrl":"https://doi.org/10.1214/22-aap1781","url":null,"abstract":"We develop a novel procedure for estimating the optimizer of general convex stochastic optimization problems of the form minx∈X E[F (x, ξ)], when the given data is a finite independent sample selected according to ξ. The procedure is based on a median-of-means tournament, and is the first procedure that exhibits the optimal statistical performance in heavy tailed situations: we recover the asymptotic rates dictated by the central limit theorem in a non-asymptotic manner once the sample size exceeds some explicitly computable threshold. Additionally, our results apply in the high-dimensional setup, as the threshold sample size exhibits the optimal dependence on the dimension (up to a logarithmic factor). The general setting allows us to recover recent results on multivariate mean estimation and linear regression in heavy-tailed situations and to prove the first sharp, non-asymptotic results for the portfolio optimization problem.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45600868","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 confirm a long-standing conjecture concerning shear-induced chaos in stochastically perturbed systems exhibiting a Hopf bifurcation. The method of showing the main chaotic property, a positive Lyapunov exponent, is a computer-assisted proof. Using the recently developed theory of conditioned Lyapunov exponents on bounded domains and the modified Furstenberg-Khasminskii formula, the problem boils down to the rigorous computation of eigenfunctions of the Kolmogorov operators describing distributions of the underlying stochastic process.
{"title":"Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems","authors":"M. Breden, Maximilian Engel","doi":"10.1214/22-aap1841","DOIUrl":"https://doi.org/10.1214/22-aap1841","url":null,"abstract":"We confirm a long-standing conjecture concerning shear-induced chaos in stochastically perturbed systems exhibiting a Hopf bifurcation. The method of showing the main chaotic property, a positive Lyapunov exponent, is a computer-assisted proof. Using the recently developed theory of conditioned Lyapunov exponents on bounded domains and the modified Furstenberg-Khasminskii formula, the problem boils down to the rigorous computation of eigenfunctions of the Kolmogorov operators describing distributions of the underlying stochastic process.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66087832","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}
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of weights, the first order for the height of such trees has been recently established by one of the authors. In this paper, we obtain the second and third orders in the asymptotic expansion of the height of weighted recursive trees, under similar assumptions. Our methods are inspired from those used to prove similar results for branching random walks. Our results also apply to a related model of growing trees, called the preferential attachment tree with additive fitnesses.
{"title":"Correction terms for the height of weighted recursive trees","authors":"Michel Pain, Delphin S'enizergues","doi":"10.1214/21-aap1756","DOIUrl":"https://doi.org/10.1214/21-aap1756","url":null,"abstract":"Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of weights, the first order for the height of such trees has been recently established by one of the authors. In this paper, we obtain the second and third orders in the asymptotic expansion of the height of weighted recursive trees, under similar assumptions. Our methods are inspired from those used to prove similar results for branching random walks. Our results also apply to a related model of growing trees, called the preferential attachment tree with additive fitnesses.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45519346","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}
Andrea Cosso, Fausto Gozzi, Idris Kharroubi, H. Pham, M. Rosestolato
We study the optimal control of path-dependent McKean-Vlasov equations valued in Hilbert spaces motivated by non Markovian mean-field models driven by stochastic PDEs. We first establish the well-posedness of the state equation, and then we prove the dynamic programming principle (DPP) in such a general framework. The crucial law invariance property of the value function V is rigorously obtained, which means that V can be viewed as a function on the Wasserstein space of probability measures on the set of continuous functions valued in Hilbert space. We then define a notion of pathwise measure derivative, which extends the Wasserstein derivative due to Lions [41], and prove a related functional It{^o} formula in the spirit of Dupire [24] and Wu and Zhang [51]. The Master Bellman equation is derived from the DPP by means of a suitable notion of viscosity solution. We provide different formulations and simplifications of such a Bellman equation notably in the special case when there is no dependence on the law of the control.
研究了由随机偏微分方程驱动的非马尔可夫平均场模型驱动的Hilbert空间中路径相关McKean-Vlasov方程的最优控制问题。首先建立了状态方程的适定性，然后在此一般框架下证明了动态规划原理。严格地得到了值函数V的关键律不变性质，这意味着V可以看作是Hilbert空间中连续函数集合上的概率测度的Wasserstein空间上的函数。然后，我们定义了路径测度导数的概念，将Wasserstein导数推广到Lions[41]，并在Dupire[24]和Wu and Zhang[51]的精神下证明了一个相关的泛函It{^o}公式。利用合适的粘度解的概念，从DPP推导出主贝尔曼方程。我们提供了这样一个Bellman方程的不同的公式和简化，特别是在不依赖于控制律的特殊情况下。
{"title":"Optimal control of path-dependent McKean–Vlasov SDEs in infinite-dimension","authors":"Andrea Cosso, Fausto Gozzi, Idris Kharroubi, H. Pham, M. Rosestolato","doi":"10.1214/22-aap1880","DOIUrl":"https://doi.org/10.1214/22-aap1880","url":null,"abstract":"We study the optimal control of path-dependent McKean-Vlasov equations valued in Hilbert spaces motivated by non Markovian mean-field models driven by stochastic PDEs. We first establish the well-posedness of the state equation, and then we prove the dynamic programming principle (DPP) in such a general framework. The crucial law invariance property of the value function V is rigorously obtained, which means that V can be viewed as a function on the Wasserstein space of probability measures on the set of continuous functions valued in Hilbert space. We then define a notion of pathwise measure derivative, which extends the Wasserstein derivative due to Lions [41], and prove a related functional It{^o} formula in the spirit of Dupire [24] and Wu and Zhang [51]. The Master Bellman equation is derived from the DPP by means of a suitable notion of viscosity solution. We provide different formulations and simplifications of such a Bellman equation notably in the special case when there is no dependence on the law of the control.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48567057","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 focus on non-asymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (non-constant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step ({usually called Unadjusted Langevin Algorithm in the Monte Carlo literature}) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under {uniform} ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for Total Variation and $L^1$-Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using {Stochastic Analysis} adapted to decreasing step setting.
{"title":"Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds","authors":"G. Pagès, Fabien Panloup","doi":"10.1214/22-aap1828","DOIUrl":"https://doi.org/10.1214/22-aap1828","url":null,"abstract":"In this paper, we focus on non-asymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (non-constant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step ({usually called Unadjusted Langevin Algorithm in the Monte Carlo literature}) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under {uniform} ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for Total Variation and $L^1$-Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using {Stochastic Analysis} adapted to decreasing step setting.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46885229","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}
Giorgio Cipolloni, L'aszl'o ErdHos, Dominik Schroder
We consider the fluctuations of regular functions $f$ of a Wigner matrix $W$ viewed as an entire matrix $f(W)$. Going beyond the well studied tracial mode, $mathrm{Tr}[f(W)]$, which is equivalent to the customary linear statistics of eigenvalues, we show that $mathrm{Tr}[f(W)]$ is asymptotically normal for any non-trivial bounded deterministic matrix $A$. We identify three different and asymptotically independent modes of this fluctuation, corresponding to the tracial part, the traceless diagonal part and the off-diagonal part of $f(W)$ in the entire mesoscopic regime, where we find that the off-diagonal modes fluctuate on a much smaller scale than the tracial mode. In addition, we determine the fluctuations in the Eigenstate Thermalisation Hypothesis [Deutsch 1991], i.e. prove that the eigenfunction overlaps with any deterministic matrix are asymptotically Gaussian after a small spectral averaging. In particular, in the macroscopic regime our result generalises [Lytova 2013] to complex $W$ and to all crossover ensembles in between. The main technical inputs are the recent multi-resolvent local laws with traceless deterministic matrices from the companion paper [Cipolloni, ErdH{o}s, Schr"oder 2020].
{"title":"Functional central limit theorems for Wigner matrices","authors":"Giorgio Cipolloni, L'aszl'o ErdHos, Dominik Schroder","doi":"10.1214/22-aap1820","DOIUrl":"https://doi.org/10.1214/22-aap1820","url":null,"abstract":"We consider the fluctuations of regular functions $f$ of a Wigner matrix $W$ viewed as an entire matrix $f(W)$. Going beyond the well studied tracial mode, $mathrm{Tr}[f(W)]$, which is equivalent to the customary linear statistics of eigenvalues, we show that $mathrm{Tr}[f(W)]$ is asymptotically normal for any non-trivial bounded deterministic matrix $A$. We identify three different and asymptotically independent modes of this fluctuation, corresponding to the tracial part, the traceless diagonal part and the off-diagonal part of $f(W)$ in the entire mesoscopic regime, where we find that the off-diagonal modes fluctuate on a much smaller scale than the tracial mode. In addition, we determine the fluctuations in the Eigenstate Thermalisation Hypothesis [Deutsch 1991], i.e. prove that the eigenfunction overlaps with any deterministic matrix are asymptotically Gaussian after a small spectral averaging. In particular, in the macroscopic regime our result generalises [Lytova 2013] to complex $W$ and to all crossover ensembles in between. The main technical inputs are the recent multi-resolvent local laws with traceless deterministic matrices from the companion paper [Cipolloni, ErdH{o}s, Schr\"oder 2020].","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43322804","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}
Pub Date : 2020-12-17DOI: 10.47004/wp.cem.2021.0821
V. Chernozhukov, D. Chetverikov, Yuta Koike
In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of $n$ independent high-dimensional centered random vectors $X_1,dots,X_n$ over the class of rectangles in the case when the covariance matrix of the scaled average is non-degenerate. In the case of bounded $X_i$'s, the implied bound for the Kolmogorov distance between the distribution of the scaled average and the Gaussian vector takes the form $$C (B^2_n log^3 d/n)^{1/2} log n,$$ where $d$ is the dimension of the vectors and $B_n$ is a uniform envelope constant on components of $X_i$'s. This bound is sharp in terms of $d$ and $B_n$, and is nearly (up to $log n$) sharp in terms of the sample size $n$. In addition, we show that similar bounds hold for the multiplier and empirical bootstrap approximations. Moreover, we establish bounds that allow for unbounded $X_i$'s, formulated solely in terms of moments of $X_i$'s. Finally, we demonstrate that the bounds can be further improved in some special smooth and zero-skewness cases.
{"title":"Nearly optimal central limit theorem and bootstrap approximations in high dimensions","authors":"V. Chernozhukov, D. Chetverikov, Yuta Koike","doi":"10.47004/wp.cem.2021.0821","DOIUrl":"https://doi.org/10.47004/wp.cem.2021.0821","url":null,"abstract":"In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of $n$ independent high-dimensional centered random vectors $X_1,dots,X_n$ over the class of rectangles in the case when the covariance matrix of the scaled average is non-degenerate. In the case of bounded $X_i$'s, the implied bound for the Kolmogorov distance between the distribution of the scaled average and the Gaussian vector takes the form $$C (B^2_n log^3 d/n)^{1/2} log n,$$ where $d$ is the dimension of the vectors and $B_n$ is a uniform envelope constant on components of $X_i$'s. This bound is sharp in terms of $d$ and $B_n$, and is nearly (up to $log n$) sharp in terms of the sample size $n$. In addition, we show that similar bounds hold for the multiplier and empirical bootstrap approximations. Moreover, we establish bounds that allow for unbounded $X_i$'s, formulated solely in terms of moments of $X_i$'s. Finally, we demonstrate that the bounds can be further improved in some special smooth and zero-skewness cases.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44169995","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 an averaging principle for interacting slow-fast systems driven by independent fractional Brownian motions. The mode of convergence is in H"older norm in probability. We also establish geometric ergodicity for a class of fractional-driven stochastic differential equations, partially improving a recent result of Panloup and Richard.
{"title":"Slow-fast systems with fractional environment and dynamics","authors":"Xue-Mei Li, J. Sieber","doi":"10.1214/22-AAP1779","DOIUrl":"https://doi.org/10.1214/22-AAP1779","url":null,"abstract":"We prove an averaging principle for interacting slow-fast systems driven by independent fractional Brownian motions. The mode of convergence is in H\"older norm in probability. We also establish geometric ergodicity for a class of fractional-driven stochastic differential equations, partially improving a recent result of Panloup and Richard.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41773938","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 the performance of splitting algorithms, and in particular the RESTART method, for the numerical approximation of the probability that a process leaves a neighborhood of a metastable point during some long time interval [0, T ]. We show that, in contrast to alternatives such as importance sampling, the decay rate of the second moment does not degrade as T → ∞. In the course of the analysis we develop some related large deviation estimates that apply when the time interval of interest depends on the large deviation parameter.
{"title":"Splitting algorithms for rare event simulation over long time intervals","authors":"Anne Buijsrogge, P. Dupuis, M. Snarski","doi":"10.1214/20-aap1578","DOIUrl":"https://doi.org/10.1214/20-aap1578","url":null,"abstract":"In this paper we study the performance of splitting algorithms, and in particular the RESTART method, for the numerical approximation of the probability that a process leaves a neighborhood of a metastable point during some long time interval [0, T ]. We show that, in contrast to alternatives such as importance sampling, the decay rate of the second moment does not degrade as T → ∞. In the course of the analysis we develop some related large deviation estimates that apply when the time interval of interest depends on the large deviation parameter.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86702853","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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