We consider an iterated Kolmogorov diffusion X t of step n . The small ball problem for X t is solved by means of the Gaussian correlation inequality. We also prove Chung’s laws of iterated logarithm for X t both at time zero and infinity.
{"title":"An application of the Gaussian correlation inequality to the small deviations for a Kolmogorov diffusion","authors":"M. Carfagnini","doi":"10.1214/22-ecp459","DOIUrl":"https://doi.org/10.1214/22-ecp459","url":null,"abstract":"We consider an iterated Kolmogorov diffusion X t of step n . The small ball problem for X t is solved by means of the Gaussian correlation inequality. We also prove Chung’s laws of iterated logarithm for X t both at time zero and infinity.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48198342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the limiting spectral distribution of large-dimensional sample covariance matrices associated with symmetric random tensors formed by $binom{n}{d}$ different products of $d$ variables chosen from $n$ independent standardized random variables. We find optimal sufficient conditions for this distribution to be the Marchenko-Pastur law in the case $d=d(n)$ and $ntoinfty$. Our conditions reduce to $d^2=o(n)$ when the variables have uniformly bounded fourth moments. The proofs are based on a new concentration inequality for quadratic forms in symmetric random tensors and a law of large numbers for elementary symmetric random polynomials.
{"title":"Marchenko-Pastur law for a random tensor model","authors":"P. Yaskov","doi":"10.1214/23-ecp527","DOIUrl":"https://doi.org/10.1214/23-ecp527","url":null,"abstract":"We study the limiting spectral distribution of large-dimensional sample covariance matrices associated with symmetric random tensors formed by $binom{n}{d}$ different products of $d$ variables chosen from $n$ independent standardized random variables. We find optimal sufficient conditions for this distribution to be the Marchenko-Pastur law in the case $d=d(n)$ and $ntoinfty$. Our conditions reduce to $d^2=o(n)$ when the variables have uniformly bounded fourth moments. The proofs are based on a new concentration inequality for quadratic forms in symmetric random tensors and a law of large numbers for elementary symmetric random polynomials.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44325550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that for low enough temperatures, but still above the AT line, the Jacobian of the TAP equations for the SK model has a macroscopic fraction of eigenvalues outside the unit interval. This provides a simple explanation for the numerical instability of the fixed points, which thus occurs already in high temperature. The insight leads to some algorithmic considerations on the low temperature regime, also briefly discussed.
{"title":"TAP equations are repulsive","authors":"Stephan Gufler, Jan Lukas Igelbrink, N. Kistler","doi":"10.1214/22-ecp505","DOIUrl":"https://doi.org/10.1214/22-ecp505","url":null,"abstract":"We show that for low enough temperatures, but still above the AT line, the Jacobian of the TAP equations for the SK model has a macroscopic fraction of eigenvalues outside the unit interval. This provides a simple explanation for the numerical instability of the fixed points, which thus occurs already in high temperature. The insight leads to some algorithmic considerations on the low temperature regime, also briefly discussed.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43099481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper concerns a variational representation formula for Wiener functionals. Let B = { B t } t ≥ 0 be a standard d -dimensional Brownian motion. Boué and Dupuis (1998) showed that, for any bounded measurable functional F ( B ) of B up to time 1 , the expectation E (cid:104) e F ( B ) (cid:105) admits a variational representation in terms of drifted Brownian motions. In this paper, with a slight modification of insightful reasoning by Lehec (2013) allowing also F ( B ) to be a functional of B over the whole time interval, we prove that the Boué–Dupuis formula holds true provided that both e F ( B ) and F ( B ) are integrable, relaxing conditions in earlier works. We also show that the formula implies the exponential hypercontractivity of the Ornstein–Uhlenbeck semigroup in R d , and hence, due to their equivalence, implies the logarithmic Sobolev inequality in the d -dimensional Gaussian space.
本文讨论了维纳泛函的一个变分表示公式。设B = {B t} t≥0为标准d维布朗运动。bouboure和Dupuis(1998)表明，对于任何有界的可测泛函F (B)，对于时间1，期望E (cid:104) E F (B) (cid:105)允许在漂移布朗运动方面的变分表示。在本文中，对Lehec(2013)的深刻推理稍加修改，允许F (B)也是B在整个时间区间内的泛函，我们证明了在e F (B)和F (B)都是可积的松弛条件下，bou - dupuis公式成立。我们还证明了该公式暗示了R d中Ornstein-Uhlenbeck半群的指数超收缩性，因此，由于它们的等价性，暗示了d维高斯空间中的对数Sobolev不等式。
{"title":"The Boué–Dupuis formula and the exponential hypercontractivity in the Gaussian space","authors":"Yuu Hariya, S. Watanabe","doi":"10.1214/22-ECP461","DOIUrl":"https://doi.org/10.1214/22-ECP461","url":null,"abstract":"This paper concerns a variational representation formula for Wiener functionals. Let B = { B t } t ≥ 0 be a standard d -dimensional Brownian motion. Boué and Dupuis (1998) showed that, for any bounded measurable functional F ( B ) of B up to time 1 , the expectation E (cid:104) e F ( B ) (cid:105) admits a variational representation in terms of drifted Brownian motions. In this paper, with a slight modification of insightful reasoning by Lehec (2013) allowing also F ( B ) to be a functional of B over the whole time interval, we prove that the Boué–Dupuis formula holds true provided that both e F ( B ) and F ( B ) are integrable, relaxing conditions in earlier works. We also show that the formula implies the exponential hypercontractivity of the Ornstein–Uhlenbeck semigroup in R d , and hence, due to their equivalence, implies the logarithmic Sobolev inequality in the d -dimensional Gaussian space.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41409898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove asymptotic formulas for the expectation of the vertex number and missed area of uniform random disc-polygons in convex disc-polygons. Our statements are the $r$-convex analogues of the classical results of R'enyi and Sulanke (1964) about random polygons in convex polygons.
{"title":"On random disc-polygons in a disc-polygon","authors":"F. Fodor, P. Kevei, V. V'igh","doi":"10.1214/23-ecp515","DOIUrl":"https://doi.org/10.1214/23-ecp515","url":null,"abstract":"We prove asymptotic formulas for the expectation of the vertex number and missed area of uniform random disc-polygons in convex disc-polygons. Our statements are the $r$-convex analogues of the classical results of R'enyi and Sulanke (1964) about random polygons in convex polygons.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42366962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we extend some recent results on systems of backward stochastic differential equations (BSDEs) with quadratic growth to the case of coupled forward-backward stochastic differential equations (FBSDEs). We work in a Markovian setting, and use results from the quadratic BSDE literature together with PDE techniques to obtain a-priori estimates which lead to an existence result. We also identify a general class of stochastic differential games whose corresponding FBSDE systems are covered by our main existence result. This leads to the existence of Markovian Nash equilibria for such games.
{"title":"Global existence for quadratic FBSDE systems and application to stochastic differential games","authors":"Joe Jackson","doi":"10.1214/23-ecp513","DOIUrl":"https://doi.org/10.1214/23-ecp513","url":null,"abstract":"In this note, we extend some recent results on systems of backward stochastic differential equations (BSDEs) with quadratic growth to the case of coupled forward-backward stochastic differential equations (FBSDEs). We work in a Markovian setting, and use results from the quadratic BSDE literature together with PDE techniques to obtain a-priori estimates which lead to an existence result. We also identify a general class of stochastic differential games whose corresponding FBSDE systems are covered by our main existence result. This leads to the existence of Markovian Nash equilibria for such games.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48112569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we establish that the stationary distribution of a possibly non-equilibrium Langevin diffusion converges, as the damping parameter goes to infinity (or equiva-lently in the Smoluchowski-Kramers vanishing mass limit), toward a tensor product of the stationary distribution of the corresponding overdamped process and of a Gaussian distribution.
{"title":"Overdamped limit at stationarity for non-equilibrium Langevin diffusions","authors":"Pierre Monmarch'e, M. Ramil","doi":"10.1214/22-ecp447","DOIUrl":"https://doi.org/10.1214/22-ecp447","url":null,"abstract":"In this note, we establish that the stationary distribution of a possibly non-equilibrium Langevin diffusion converges, as the damping parameter goes to infinity (or equiva-lently in the Smoluchowski-Kramers vanishing mass limit), toward a tensor product of the stationary distribution of the corresponding overdamped process and of a Gaussian distribution.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45289593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Based on two isomorphisms of Hopf algebras, we provide a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths.
基于Hopf代数的两个同构，给出了分支粗糙路径驱动的受控微分方程截断泰勒展开余数的最优阶界。
{"title":"A remainder estimate for branched rough differential equations","authors":"Danyu Yang","doi":"10.1214/22-ECP473","DOIUrl":"https://doi.org/10.1214/22-ECP473","url":null,"abstract":"Based on two isomorphisms of Hopf algebras, we provide a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43070543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 extend upon a result by Mueller and Tribe regarding Funaki’s model of a random string. Specifically, we examine the rate of escape of this model in dimensions d ≥ 7 . We also provide a bound for the rate of approach to the origin in dimension d = 6 .
{"title":"On the rate of escape or approach to the origin of a random string","authors":"P. Lam","doi":"10.1214/22-ecp451","DOIUrl":"https://doi.org/10.1214/22-ecp451","url":null,"abstract":". In this paper, we extend upon a result by Mueller and Tribe regarding Funaki’s model of a random string. Specifically, we examine the rate of escape of this model in dimensions d ≥ 7 . We also provide a bound for the rate of approach to the origin in dimension d = 6 .","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48886709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the eigenvalue trajectories of a time dependent matrix $ G_t = H+i t vv^*$ for $t geq 0$, where $H$ is an $N times N$ Hermitian random matrix and $v$ is a unit vector. In particular, we establish that with high probability, an outlier can be distinguished at all times $t>1+N^{-1/3+epsilon}$, for any $epsilon>0$. The study of this natural process combines elements of Hermitian and non-Hermitian analysis, and illustrates some aspects of the intrinsic instability of (even weakly) non-Hermitian matrices.
{"title":"Dynamics of a rank-one perturbation of a Hermitian matrix","authors":"Guillaume Dubach, L'aszl'o ErdHos","doi":"10.1214/23-ECP516","DOIUrl":"https://doi.org/10.1214/23-ECP516","url":null,"abstract":"We study the eigenvalue trajectories of a time dependent matrix $ G_t = H+i t vv^*$ for $t geq 0$, where $H$ is an $N times N$ Hermitian random matrix and $v$ is a unit vector. In particular, we establish that with high probability, an outlier can be distinguished at all times $t>1+N^{-1/3+epsilon}$, for any $epsilon>0$. The study of this natural process combines elements of Hermitian and non-Hermitian analysis, and illustrates some aspects of the intrinsic instability of (even weakly) non-Hermitian matrices.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47386029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：