We present a large deviation theory that characterizes the exponential estimate for rare events in stochastic dynamical systems in the limit of weak noise.We aim to consider a next-to-leading-order approximation for m...We present a large deviation theory that characterizes the exponential estimate for rare events in stochastic dynamical systems in the limit of weak noise.We aim to consider a next-to-leading-order approximation for more accurate calculation of the mean exit time by computing large deviation prefactors with the aid of machine learning.More specifically,we design a neural network framework to compute quasipotential,most probable paths and prefactors based on the orthogonal decomposition of a vector field.We corroborate the higher effectiveness and accuracy of our algorithm with two toy models.Numerical experiments demonstrate its powerful functionality in exploring the internal mechanism of rare events triggered by weak random fluctuations.展开更多
Letλ=(λ_(1),…,λ_(n))beβ-Jacobi ensembles with parameters p_(1),p_(2),n andβ,withβvarying with n.Set■.Suppose that■and 0≤σγ<1.We offer the large deviation for p_(1)+p_(2)/p_(1)■λ_(i)whenγ>0 via the...Letλ=(λ_(1),…,λ_(n))beβ-Jacobi ensembles with parameters p_(1),p_(2),n andβ,withβvarying with n.Set■.Suppose that■and 0≤σγ<1.We offer the large deviation for p_(1)+p_(2)/p_(1)■λ_(i)whenγ>0 via the large deviation of the corresponding empirical measure and via a direct estimate,respectively,whenγ=0.展开更多
In this paper,we establish a large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise.The main difficulties come from the highly non-linear coefficient and the jump noise....In this paper,we establish a large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise.The main difficulties come from the highly non-linear coefficient and the jump noise.Here,we adopt a new sufficient condition for the weak convergence criterion of the large deviation principle,which was initially proposed by Matoussi,Sabbagh and Zhang(2021).展开更多
M-negatively associated random variables, which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its particular case, are introduced and studied. Large devi...M-negatively associated random variables, which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its particular case, are introduced and studied. Large deviation principles and moderate deviation upper bounds for stationary m-negatively associated random variables are proved. Kolmogorov-type and Marcinkiewicz-type strong laws of large numbers as well as the three series theorem for m-negatively associated random variables are also given.展开更多
Let (Xi) be a martingale difference sequence and Sn=∑^ni=1Xi Suppose (Xi) i=1 is bounded in L^p. In the case p ≥2, Lesigne and Volny (Stochastic Process. Appl. 96 (2001) 143) obtained the estimation μ(Sn ...Let (Xi) be a martingale difference sequence and Sn=∑^ni=1Xi Suppose (Xi) i=1 is bounded in L^p. In the case p ≥2, Lesigne and Volny (Stochastic Process. Appl. 96 (2001) 143) obtained the estimation μ(Sn 〉 n) ≤ cn^-p/2, Yulin Li (Statist. Probab. Lett. 62 (2003) 317) generalized the result to the case when p ∈ (1,2] and obtained μ(Sn 〉 n) ≤ cn^l-p, these are optimal in a certain sense. In this article, the authors study the large deviation of Sn for some dependent sequences and obtain the same order optimal upper bounds for μ(Sn 〉 n) as those for martingale difference sequence.展开更多
We give a definition of relative entropy with respect to a sublinear expectation and establish large deviation principle for the empirical measures for independent random variables under the sublinear expectation.
Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) ...Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) and describe the decay rates of the n-step transition probabilities.As applications,a large deviation principle for the sequence log Z_(n) is established,and related large deviations are also studied.展开更多
In this article, the author obtains the large deviation principles for the empirical correlation coefficient of two Gaussian random variables X and Y. Especially, when considering two independent Gaussian random varia...In this article, the author obtains the large deviation principles for the empirical correlation coefficient of two Gaussian random variables X and Y. Especially, when considering two independent Gaussian random variables X, Y with the means EX, EY (both known), wherein the author gives two kinds of different proofs and gets the same results.展开更多
In this article, we obtain the large deviations and moderate deviations for negatively dependent (ND) and non-identically distributed random variables defined on (-∞, +∞). The results show that for some non-ide...In this article, we obtain the large deviations and moderate deviations for negatively dependent (ND) and non-identically distributed random variables defined on (-∞, +∞). The results show that for some non-identical random variables, precise large deviations and moderate deviations remain insensitive to negative dependence structure.展开更多
Consider a multidimensional renewal risk model, in which the claim sizes {Xk, k ≥1} form a sequence of independent and identically distributed random vectors with nonnegative components that are allowed to be depende...Consider a multidimensional renewal risk model, in which the claim sizes {Xk, k ≥1} form a sequence of independent and identically distributed random vectors with nonnegative components that are allowed to be dependent on each other. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Suppose that the claim sizes and inter-arrival times correspondingly form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure. A precise large deviation for the multidimensional renewal risk model is obtained.展开更多
In this article, we prove upper large deviations for the empirical measure generated by stationary mixing random sequence under some suitable assumptions and upper large deviations for the mixing random sequence.
In this paper, we study the precise large deviations for the prospectiveloss process with consistently varying tails. The obtained results improve some related known ones.
In this paper the large deviation results for partial and random sums Sn-ESn=n∑i=1Xi-n∑i=1EXi,n≥1;S(t)-ES(t)=N(t)∑i=1Xi-E(N(t)∑i=1Xi),t≥0 are proved, where {N(t);t ≥ 0} is a counting process of non-...In this paper the large deviation results for partial and random sums Sn-ESn=n∑i=1Xi-n∑i=1EXi,n≥1;S(t)-ES(t)=N(t)∑i=1Xi-E(N(t)∑i=1Xi),t≥0 are proved, where {N(t);t ≥ 0} is a counting process of non-negative integer-valued random variables, and {Xn; n ≥ 1} are a sequence of independent non-negative random variables independent of {N(t); t ≥ 0}. These results extend and improve some known conclusions.展开更多
We establish a Freidlin-Wentzell’s large deviation principle for general stochastic evolution equations with Poisson jumps and small multiplicative noises by using weak convergence method.
We study the asymptotics tot the statistic of chi-square in type Ⅱ error. By the contraction principle, the large deviations and moderate deviations are obtained, and the rate function of moderate deviations can be c...We study the asymptotics tot the statistic of chi-square in type Ⅱ error. By the contraction principle, the large deviations and moderate deviations are obtained, and the rate function of moderate deviations can be calculated explicitly which is a squared function.展开更多
Let u ∈ R ,for any ω 〉 0, the processes X^ε = {X^ε(t); 0 ≤ t≤ 1} are governed by the following random evolution equations dX^ε(t)= b(X^ε(t),v(t))dt-εdSt/ε, where S={St; 0≤t≤1} is a compound Pois...Let u ∈ R ,for any ω 〉 0, the processes X^ε = {X^ε(t); 0 ≤ t≤ 1} are governed by the following random evolution equations dX^ε(t)= b(X^ε(t),v(t))dt-εdSt/ε, where S={St; 0≤t≤1} is a compound Poisson process, the process v={v(t); 0≤t≤1} is independent of S and takes values in R^m. We derive the large deviation principle for{(X^ε,v(.)); ε〉0} when ε↓0 by approximation method and contraction principle, which will be meaningful for us to find out the path property for the risk process of this type.展开更多
By the Cramér method, the large deviation principle for a form of compound Poisson process S(t)=∑N(t)i=1h(t-Si)Xi is obtained,where N(t), t>0, is a nonhomogeneous Poisson process with intensity λ(t)>0, Xi...By the Cramér method, the large deviation principle for a form of compound Poisson process S(t)=∑N(t)i=1h(t-Si)Xi is obtained,where N(t), t>0, is a nonhomogeneous Poisson process with intensity λ(t)>0, Xi, i≥1, are i.i.d. nonnegative random variables independent of N(t), and h(t), t>0, is a nonnegative monotone real function. Consequently, weak convergence for S(t) is also obtained.展开更多
Letλ=(λ_(1),...,λ_(n))beβ-Jacobi ensembles with parameters p_(1),p_(2),n andβwhileβvarying with n.Setγ=lim_(n→∞)n/p_(1)andσ=lim_(n→∞)p_(1)/p_(2).In this paper,supposing lim_(n→∞)log_(n)/β_(n)=0,we prove...Letλ=(λ_(1),...,λ_(n))beβ-Jacobi ensembles with parameters p_(1),p_(2),n andβwhileβvarying with n.Setγ=lim_(n→∞)n/p_(1)andσ=lim_(n→∞)p_(1)/p_(2).In this paper,supposing lim_(n→∞)log_(n)/β_(n)=0,we prove that the empirical measures of different scaledλconverge weakly to a Wachter distribution,a Marchenko–Pastur law and a semicircle law corresponding toσγ>0,σ=0 orγ=0,respectively.We also offer a full large deviation principle with speedβn^(2)and a good rate function to precise the speed of these convergences.As an application,the strong law of large numbers for the extremal eigenvalues ofβ-Jacobi ensembles is obtained.展开更多
Let(λ_(1),...,λ_(n)) be β-Laguerre ensembles with parametersβ,p,n and μ_(n):=1/n∑^(n)_(i=1)δ_(Xi) with for 1≤i≤n.In this paper,asβvaries and satisfies lim_(n→∞)log n/βn=0,we offer a modified Marchenko-Pas...Let(λ_(1),...,λ_(n)) be β-Laguerre ensembles with parametersβ,p,n and μ_(n):=1/n∑^(n)_(i=1)δ_(Xi) with for 1≤i≤n.In this paper,asβvaries and satisfies lim_(n→∞)log n/βn=0,we offer a modified Marchenko-Pastur,law or semicircle law as the weak limits for the sequence μ_(n) when lim_(n→∞)n/p=γ∈(0,1]or lim_(n→∞)n/p=0,respectively.This recovers some well-known results.Moreover,we give a full large deviation principle of μ_(n) with speed βn^(2) and good rate function I_(γ) under the same condition to characterize the speed of the convergence.The minimizer of I_(γ) is a modified Marchenko-Pastur law forγ∈(0,1]and the semicircle law forγ=0.展开更多
In this paper we establish the large deviation principle for the the two-dimensional stochastic Navier-Stokes equations with anisotropic viscosity both for small noise and for short time.The proof for large deviation ...In this paper we establish the large deviation principle for the the two-dimensional stochastic Navier-Stokes equations with anisotropic viscosity both for small noise and for short time.The proof for large deviation principle is based on the weak convergence approach.For small time asymptotics we use the exponential equivalence to prove the result.展开更多
基金Project supported by the Natural Science Foundation of Jiangsu Province (Grant No.BK20220917)the National Natural Science Foundation of China (Grant Nos.12001213 and 12302035)。
文摘We present a large deviation theory that characterizes the exponential estimate for rare events in stochastic dynamical systems in the limit of weak noise.We aim to consider a next-to-leading-order approximation for more accurate calculation of the mean exit time by computing large deviation prefactors with the aid of machine learning.More specifically,we design a neural network framework to compute quasipotential,most probable paths and prefactors based on the orthogonal decomposition of a vector field.We corroborate the higher effectiveness and accuracy of our algorithm with two toy models.Numerical experiments demonstrate its powerful functionality in exploring the internal mechanism of rare events triggered by weak random fluctuations.
基金supported by the NSFC (12171038,11871008)985 Projects.
文摘Letλ=(λ_(1),…,λ_(n))beβ-Jacobi ensembles with parameters p_(1),p_(2),n andβ,withβvarying with n.Set■.Suppose that■and 0≤σγ<1.We offer the large deviation for p_(1)+p_(2)/p_(1)■λ_(i)whenγ>0 via the large deviation of the corresponding empirical measure and via a direct estimate,respectively,whenγ=0.
基金partially supported by the National Natural Science Foundation of China(11871382,12071361)partially supported by the National Natural Science Foundation of China(11971361,11731012)。
文摘In this paper,we establish a large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise.The main difficulties come from the highly non-linear coefficient and the jump noise.Here,we adopt a new sufficient condition for the weak convergence criterion of the large deviation principle,which was initially proposed by Matoussi,Sabbagh and Zhang(2021).
基金Partly supported by the National Natural Science Foundation of China and the Ministry of Education of ChinaPartly supported by the Science and Technology Research Item of Hubei Provincial Department of Education,Jiaghan University
文摘M-negatively associated random variables, which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its particular case, are introduced and studied. Large deviation principles and moderate deviation upper bounds for stationary m-negatively associated random variables are proved. Kolmogorov-type and Marcinkiewicz-type strong laws of large numbers as well as the three series theorem for m-negatively associated random variables are also given.
基金the National Natural Science Foundation of China(10571001)the Innovation Group Foundation of Anhui University
文摘Let (Xi) be a martingale difference sequence and Sn=∑^ni=1Xi Suppose (Xi) i=1 is bounded in L^p. In the case p ≥2, Lesigne and Volny (Stochastic Process. Appl. 96 (2001) 143) obtained the estimation μ(Sn 〉 n) ≤ cn^-p/2, Yulin Li (Statist. Probab. Lett. 62 (2003) 317) generalized the result to the case when p ∈ (1,2] and obtained μ(Sn 〉 n) ≤ cn^l-p, these are optimal in a certain sense. In this article, the authors study the large deviation of Sn for some dependent sequences and obtain the same order optimal upper bounds for μ(Sn 〉 n) as those for martingale difference sequence.
基金supported by the National Natural Science Foundation of China(11171262)the Specialized Research Fund for the Doctoral Program of Higher Education (200804860048)
文摘We give a definition of relative entropy with respect to a sublinear expectation and establish large deviation principle for the empirical measures for independent random variables under the sublinear expectation.
基金partially supported by the National Nature Science Foundation of China(11601286,11501146)。
文摘Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) and describe the decay rates of the n-step transition probabilities.As applications,a large deviation principle for the sequence log Z_(n) is established,and related large deviations are also studied.
文摘In this article, the author obtains the large deviation principles for the empirical correlation coefficient of two Gaussian random variables X and Y. Especially, when considering two independent Gaussian random variables X, Y with the means EX, EY (both known), wherein the author gives two kinds of different proofs and gets the same results.
基金supported by the Youth Foundation of Hubei Province Department of Education of China (Q200710002)
文摘In this article, we obtain the large deviations and moderate deviations for negatively dependent (ND) and non-identically distributed random variables defined on (-∞, +∞). The results show that for some non-identical random variables, precise large deviations and moderate deviations remain insensitive to negative dependence structure.
基金Supported by the National Natural Science Foundation of China(Nos.11571058&11301481)Humanities and Social Science Foundation of the Ministry of Education of China(No.17YJC910007)+1 种基金Zhejiang Provincial Natural Science Foundation of China(No.LY17A010004)Fundamental Research Funds for the Central Universities(No.DUT17LK31)
文摘Consider a multidimensional renewal risk model, in which the claim sizes {Xk, k ≥1} form a sequence of independent and identically distributed random vectors with nonnegative components that are allowed to be dependent on each other. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Suppose that the claim sizes and inter-arrival times correspondingly form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure. A precise large deviation for the multidimensional renewal risk model is obtained.
基金The NSF (10571073) of China985 Program of Jilin University
文摘In this article, we prove upper large deviations for the empirical measure generated by stationary mixing random sequence under some suitable assumptions and upper large deviations for the mixing random sequence.
文摘In this paper, we study the precise large deviations for the prospectiveloss process with consistently varying tails. The obtained results improve some related known ones.
基金Supported by the Science Foundation of the Education Committee of Anhui Province(0505101).
文摘In this paper the large deviation results for partial and random sums Sn-ESn=n∑i=1Xi-n∑i=1EXi,n≥1;S(t)-ES(t)=N(t)∑i=1Xi-E(N(t)∑i=1Xi),t≥0 are proved, where {N(t);t ≥ 0} is a counting process of non-negative integer-valued random variables, and {Xn; n ≥ 1} are a sequence of independent non-negative random variables independent of {N(t); t ≥ 0}. These results extend and improve some known conclusions.
文摘We establish a Freidlin-Wentzell’s large deviation principle for general stochastic evolution equations with Poisson jumps and small multiplicative noises by using weak convergence method.
基金the National Natural Science Foundation of China (10571139)
文摘We study the asymptotics tot the statistic of chi-square in type Ⅱ error. By the contraction principle, the large deviations and moderate deviations are obtained, and the rate function of moderate deviations can be calculated explicitly which is a squared function.
基金Supported by the National Natural Science Foundation of China (70273029)
文摘Let u ∈ R ,for any ω 〉 0, the processes X^ε = {X^ε(t); 0 ≤ t≤ 1} are governed by the following random evolution equations dX^ε(t)= b(X^ε(t),v(t))dt-εdSt/ε, where S={St; 0≤t≤1} is a compound Poisson process, the process v={v(t); 0≤t≤1} is independent of S and takes values in R^m. We derive the large deviation principle for{(X^ε,v(.)); ε〉0} when ε↓0 by approximation method and contraction principle, which will be meaningful for us to find out the path property for the risk process of this type.
基金National Natural Science Foundation of China(No. 10971157)Educational Commission of Hubei Province, China(No.2004X124)
文摘By the Cramér method, the large deviation principle for a form of compound Poisson process S(t)=∑N(t)i=1h(t-Si)Xi is obtained,where N(t), t>0, is a nonhomogeneous Poisson process with intensity λ(t)>0, Xi, i≥1, are i.i.d. nonnegative random variables independent of N(t), and h(t), t>0, is a nonnegative monotone real function. Consequently, weak convergence for S(t) is also obtained.
基金Supported by NSFC(Grant Nos.12171038,11871008)985 Projects。
文摘Letλ=(λ_(1),...,λ_(n))beβ-Jacobi ensembles with parameters p_(1),p_(2),n andβwhileβvarying with n.Setγ=lim_(n→∞)n/p_(1)andσ=lim_(n→∞)p_(1)/p_(2).In this paper,supposing lim_(n→∞)log_(n)/β_(n)=0,we prove that the empirical measures of different scaledλconverge weakly to a Wachter distribution,a Marchenko–Pastur law and a semicircle law corresponding toσγ>0,σ=0 orγ=0,respectively.We also offer a full large deviation principle with speedβn^(2)and a good rate function to precise the speed of these convergences.As an application,the strong law of large numbers for the extremal eigenvalues ofβ-Jacobi ensembles is obtained.
基金Supported by NSFC(Grant Nos.12171038,11871008)the National Key R&D Program of China(Grant No.2020YFA0712900)985 Projects。
文摘Let(λ_(1),...,λ_(n)) be β-Laguerre ensembles with parametersβ,p,n and μ_(n):=1/n∑^(n)_(i=1)δ_(Xi) with for 1≤i≤n.In this paper,asβvaries and satisfies lim_(n→∞)log n/βn=0,we offer a modified Marchenko-Pastur,law or semicircle law as the weak limits for the sequence μ_(n) when lim_(n→∞)n/p=γ∈(0,1]or lim_(n→∞)n/p=0,respectively.This recovers some well-known results.Moreover,we give a full large deviation principle of μ_(n) with speed βn^(2) and good rate function I_(γ) under the same condition to characterize the speed of the convergence.The minimizer of I_(γ) is a modified Marchenko-Pastur law forγ∈(0,1]and the semicircle law forγ=0.
基金Research supported in part by NSFC(No.11771037)Key Lab of Random Complex Structures and Data Science,Chinese Academy of ScienceFinancial the DFG through the CRC 1283“Taming uncertainty and profiting from randomness and low regularity in analysis,stochastics and their applications”。
文摘In this paper we establish the large deviation principle for the the two-dimensional stochastic Navier-Stokes equations with anisotropic viscosity both for small noise and for short time.The proof for large deviation principle is based on the weak convergence approach.For small time asymptotics we use the exponential equivalence to prove the result.