Let {X,X n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set S n=n k=1X k,M n=max k≤n｜S k｜,n≥1. Suppose lim n→∞ES2 n/n=∶σ2>0 and ∞...Let {X,X n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set S n=n k=1X k,M n=max k≤n｜S k｜,n≥1. Suppose lim n→∞ES2 n/n=∶σ2>0 and ∞n=1ρ 2/d(2n)<∞, where d=2,if -1<b<0 and d>2(b+1),if b≥0. It is proved that,for any b>-1, limε0ε 2(b+1)∞n=1(loglogn)bnlognP{M n≥εσ2nloglogn}= 2(b+1)πГ(b+3/2)∞k=0(-1)k(2k+1) 2b+2,where Г(·) is a Gamma function.展开更多
In the case of Z＋^d（d ≥ 2）-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k∈ Z＋^d} i.i.d, random variables with mean 0, Sn =∑k≤nXk and Vn^2 = ∑j≤nXj^2, the precise asymptotics for ∑...In the case of Z＋^d（d ≥ 2）-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k∈ Z＋^d} i.i.d, random variables with mean 0, Sn =∑k≤nXk and Vn^2 = ∑j≤nXj^2, the precise asymptotics for ∑n1/｜n｜（log｜n｜dP（｜Sn/Vn｜≥ε√log log｜n｜） and ∑n（logn｜）b/｜n｜（log｜n｜）^d-1P（｜Sn/Vn｜≥ε√log n）,as ε↓0,is established.展开更多
In this article, a law of iterated logarithm for the maximum likelihood estimator in a random censoring model with incomplete information under certain regular conditions is obtained.
Let{Xn;n≥1}be a sequence of i.i.d, random variables with finite variance,Q（n）be the related R/S statistics. It is proved that lim ε↓0 ε^2 ∑n=1 ^8 n log n/1 P{Q（n）≥ε√2n log log n}=2/1 EY^2,where Y=sup0≤t...Let{Xn;n≥1}be a sequence of i.i.d, random variables with finite variance,Q（n）be the related R/S statistics. It is proved that lim ε↓0 ε^2 ∑n=1 ^8 n log n/1 P{Q（n）≥ε√2n log log n}=2/1 EY^2,where Y=sup0≤t≤1B（t）-inf0≤t≤sB（t）,and B（t） is a Brownian bridge.展开更多
Let {X, X_n, n ≥ 1} be a sequence of i.i.d. random vectors with EX =（0,..., 0）_（m×1） and Cov（X, X） = σ~2 Ⅰ_m, and set S_n =∑_（i=1）~n X_i, n ≥ 1. For every d 〉 0 and a_n =o（（log log n）^（-d））, t...Let {X, X_n, n ≥ 1} be a sequence of i.i.d. random vectors with EX =（0,..., 0）_（m×1） and Cov（X, X） = σ~2 Ⅰ_m, and set S_n =∑_（i=1）~n X_i, n ≥ 1. For every d 〉 0 and a_n =o（（log log n）^（-d））, the article deals with the precise rates in the genenralized law of the iterated logarithm for a kind of weighted infinite series of P（｜S_n｜ ≥（ε ＋ a_n）σn^（1/2）（log log n）~d）.展开更多
A nonclassical law of iterated logarithm that holds for a stationary negatively associated sequence of random variables with finite variance is proved in this paper. The proof is based on a Rosenthal type maximal ineq...A nonclassical law of iterated logarithm that holds for a stationary negatively associated sequence of random variables with finite variance is proved in this paper. The proof is based on a Rosenthal type maximal inequality and the subsequence method.This result extends the work of Klesov,Rosalsky (2001) and Shao,Su (1999).展开更多
Let u(t,x)be the solution to the one-dimensional nonlinear stochastic heat equation driven by space-time white noise with u(0,x)=1 for all x∈R.In this paper,we prove the law of the iterated logarithm(LIL for short)an...Let u(t,x)be the solution to the one-dimensional nonlinear stochastic heat equation driven by space-time white noise with u(0,x)=1 for all x∈R.In this paper,we prove the law of the iterated logarithm(LIL for short)and the functional LIL for a linear additive functional of the form∫[0,R]u(t,x)dx and the nonlinear additive functionals of the form∫[0,R]g(u(t,x))dx,where g:R→R is nonrandom and Lipschitz continuous,as R→∞for fixed t>0,using the localization argument.展开更多
Hu Shuhe gets a sufficient condition on the law of the iterated logarithm for the sums of φ-mixing sequences with duple suffixes. This paper greatly improves his condition.
Let X be a d-dimensional random vector with unknown density function f(z) = f (z1, ..., z(d)), and let f(n) be teh nearest neighbor estimator of f proposed by Loftsgaarden and Quesenberry (1965). In this paper, we est...Let X be a d-dimensional random vector with unknown density function f(z) = f (z1, ..., z(d)), and let f(n) be teh nearest neighbor estimator of f proposed by Loftsgaarden and Quesenberry (1965). In this paper, we established the law of the iterated logarithm of f(n) for general case of d greater-than-or-equal-to 1, which gives the exact pointwise strong convergence rate of f(n).展开更多
In this paper,we obtain the quasi sure local Chung’s functional law of the iterated logarithm for increments of a Brownian motion.As an application,a quasi sure Chung’s type functional modulus of continuity for a Br...In this paper,we obtain the quasi sure local Chung’s functional law of the iterated logarithm for increments of a Brownian motion.As an application,a quasi sure Chung’s type functional modulus of continuity for a Brownian motion is also derived.展开更多
In this paper,we present local functional law of the iterated logarithm for Cs?rg?-Révész type increments of fractional Brownian motion.The results obtained extend works of Gantert[Ann.Probab.,1993,21(2):104...In this paper,we present local functional law of the iterated logarithm for Cs?rg?-Révész type increments of fractional Brownian motion.The results obtained extend works of Gantert[Ann.Probab.,1993,21(2):1045-1049]and Monrad and Rootzén[Probab.Theory Related Fields,1995,101(2):173-192].展开更多
Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper est...Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments.For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sublinear expectation for random variables with only finite variances.展开更多
In this paper,we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space,where the random variables are not necessarily identically distribu...In this paper,we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space,where the random variables are not necessarily identically distributed.Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm.As an application,the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained.In the paper,it is also shown that if the sub-linear expectation space is rich enough,it will have no continuous capacity.The laws of the iterated logarithm are established without the assumption on the continuity of capacities.展开更多
A new Hartman-Wintner-type law of the iterated logarithm for independent random variables with mean-uncertainty under sublinear expectations is established by the martingale analogue of the Kolmogorov law of the itera...A new Hartman-Wintner-type law of the iterated logarithm for independent random variables with mean-uncertainty under sublinear expectations is established by the martingale analogue of the Kolmogorov law of the iterated logarithm in classical probability theory.展开更多
Consider a ρ-mixing sequence of identically distributed random variables with the underlying dis- tribution in the domain of attraction of the normal distribution. This paper proves that law of the iterated logarithm...Consider a ρ-mixing sequence of identically distributed random variables with the underlying dis- tribution in the domain of attraction of the normal distribution. This paper proves that law of the iterated logarithm holds for ρ-mixing sequences of random variables. Our results generalize and improve Theorems 1.2-1.3 of Qi and Cheng （1996） from the i.i.d, case to ρ-mixing sequences.展开更多
We establish the Strassen's law of the iterated logarithm(LIL for short)for independent and identically distributed random variables with E[X_(1)]=ε[X_(1)]=0 and Cv[X_(1)^(2)]<∞ under a sublinear expectation ...We establish the Strassen's law of the iterated logarithm(LIL for short)for independent and identically distributed random variables with E[X_(1)]=ε[X_(1)]=0 and Cv[X_(1)^(2)]<∞ under a sublinear expectation space with a countably sub-additive capacity V.We also show the LIL for upper capacity with σ=σ under some certain conditions.展开更多
With the help of large deviation of Brownian motion in the Hölder norm,local Strassen’s law of the iterated logarithm for increments of a Brownian motion in the Hölder norm is investigated.This paper promot...With the help of large deviation of Brownian motion in the Hölder norm,local Strassen’s law of the iterated logarithm for increments of a Brownian motion in the Hölder norm is investigated.This paper promotes the corresponding results by Gantert and Wei.展开更多
Let {β（s）,s ≥ O} be the standard Brownian motion in R^d with d ≥ 4 and let ｜Wr（t）｜ be the volume of the Wiener sausage associated with {β（s）, s ≥ O} observed until time t. Prom the central limit theorem o...Let {β（s）,s ≥ O} be the standard Brownian motion in R^d with d ≥ 4 and let ｜Wr（t）｜ be the volume of the Wiener sausage associated with {β（s）, s ≥ O} observed until time t. Prom the central limit theorem of Wiener sausage, we know that when d ≥ 4 the limit distribution is normal. In this paper, we study the laws of the iterated logarithm for ｜Wr（t）｜ - e｜Wr（t）｜ in this case.展开更多
We consider laws of iterated random walks in random environments. logarithm for one-dimensional transient A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environm...We consider laws of iterated random walks in random environments. logarithm for one-dimensional transient A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environments, including independent identically distributed environments and uniformly ergodic environments.展开更多
基金Research supported by the National Natural Science Foundation of China (1 0 0 71 0 72 )
文摘Let {X,X n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set S n=n k=1X k,M n=max k≤n｜S k｜,n≥1. Suppose lim n→∞ES2 n/n=∶σ2>0 and ∞n=1ρ 2/d(2n)<∞, where d=2,if -1<b<0 and d>2(b+1),if b≥0. It is proved that,for any b>-1, limε0ε 2(b+1)∞n=1(loglogn)bnlognP{M n≥εσ2nloglogn}= 2(b+1)πГ(b+3/2)∞k=0(-1)k(2k+1) 2b+2,where Г(·) is a Gamma function.
基金Project Supported by NSFC (10131040)SRFDP (2002335090)
文摘A law of iterated logarithm for R/S statistics with the help of the strong approximations of R/S statistics by functions of a Wiener process is shown.
文摘In the case of Z＋^d（d ≥ 2）-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k∈ Z＋^d} i.i.d, random variables with mean 0, Sn =∑k≤nXk and Vn^2 = ∑j≤nXj^2, the precise asymptotics for ∑n1/｜n｜（log｜n｜dP（｜Sn/Vn｜≥ε√log log｜n｜） and ∑n（logn｜）b/｜n｜（log｜n｜）^d-1P（｜Sn/Vn｜≥ε√log n）,as ε↓0,is established.
文摘In this article, a law of iterated logarithm for the maximum likelihood estimator in a random censoring model with incomplete information under certain regular conditions is obtained.
文摘Let{Xn;n≥1}be a sequence of i.i.d, random variables with finite variance,Q（n）be the related R/S statistics. It is proved that lim ε↓0 ε^2 ∑n=1 ^8 n log n/1 P{Q（n）≥ε√2n log log n}=2/1 EY^2,where Y=sup0≤t≤1B（t）-inf0≤t≤sB（t）,and B（t） is a Brownian bridge.
基金Supported by the National Natural Science Foundation of China(Grant No.61662037)the Scientific Program of Department of Education of Jiangxi Province(Grant Nos.GJJ150894GJJ150905)
文摘Let {X, X_n, n ≥ 1} be a sequence of i.i.d. random vectors with EX =（0,..., 0）_（m×1） and Cov（X, X） = σ~2 Ⅰ_m, and set S_n =∑_（i=1）~n X_i, n ≥ 1. For every d 〉 0 and a_n =o（（log log n）^（-d））, the article deals with the precise rates in the genenralized law of the iterated logarithm for a kind of weighted infinite series of P（｜S_n｜ ≥（ε ＋ a_n）σn^（1/2）（log log n）~d）.
文摘A nonclassical law of iterated logarithm that holds for a stationary negatively associated sequence of random variables with finite variance is proved in this paper. The proof is based on a Rosenthal type maximal inequality and the subsequence method.This result extends the work of Klesov,Rosalsky (2001) and Shao,Su (1999).
基金supported by the National Natural Science Foundation of China(11771178 and 12171198)the Science and Technology Development Program of Jilin Province(20210101467JC)+1 种基金the Science and Technology Program of Jilin Educational Department during the“13th Five-Year”Plan Period(JJKH20200951KJ)the Fundamental Research Funds for the Central Universities。
文摘Let u(t,x)be the solution to the one-dimensional nonlinear stochastic heat equation driven by space-time white noise with u(0,x)=1 for all x∈R.In this paper,we prove the law of the iterated logarithm(LIL for short)and the functional LIL for a linear additive functional of the form∫[0,R]u(t,x)dx and the nonlinear additive functionals of the form∫[0,R]g(u(t,x))dx,where g:R→R is nonrandom and Lipschitz continuous,as R→∞for fixed t>0,using the localization argument.
文摘Hu Shuhe gets a sufficient condition on the law of the iterated logarithm for the sums of φ-mixing sequences with duple suffixes. This paper greatly improves his condition.
基金Research supported by National Natural Science Foundation of China.
文摘Let X be a d-dimensional random vector with unknown density function f(z) = f (z1, ..., z(d)), and let f(n) be teh nearest neighbor estimator of f proposed by Loftsgaarden and Quesenberry (1965). In this paper, we established the law of the iterated logarithm of f(n) for general case of d greater-than-or-equal-to 1, which gives the exact pointwise strong convergence rate of f(n).
基金Supported by National Natural Science Foundation of China(11661025)Science Research Foundation of Guangxi Education Department(YB2014117)+1 种基金Guangxi Natural Science Foundation(2018GXNSFBA281076,2017GXNSFBA198179)Guangxi Programme for Promoting Young Teachers’s Ability(2018KY0214).
文摘In this paper,we obtain the quasi sure local Chung’s functional law of the iterated logarithm for increments of a Brownian motion.As an application,a quasi sure Chung’s type functional modulus of continuity for a Brownian motion is also derived.
基金Supported by NSFC(Nos.11661025,12161024)Natural Science Foundation of Guangxi(Nos.2020GXNSFAA159118,2021GXNSFAA196045)+2 种基金Guangxi Science and Technology Project(No.Guike AD20297006)Training Program for 1000 Young and Middle-aged Cadre Teachers in Universities of GuangxiNational College Student's Innovation and Entrepreneurship Training Program(No.202110595049)。
文摘In this paper,we present local functional law of the iterated logarithm for Cs?rg?-Révész type increments of fractional Brownian motion.The results obtained extend works of Gantert[Ann.Probab.,1993,21(2):1045-1049]and Monrad and Rootzén[Probab.Theory Related Fields,1995,101(2):173-192].
基金supported by National Natural Science Foundation of China (Grant No. 11225104)the National Basic Research Program of China (Grant No. 2015CB352302)the Fundamental Research Funds for the Central Universities
文摘Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments.For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sublinear expectation for random variables with only finite variances.
基金the National Natural Science Foundation of China(Grant Nos.11731012,12031005)Ten Thousand Talents Plan of Zhejiang Province(Grant No.2018R52042)+1 种基金Natural Science Foundation of Zhejiang Province(Grant No.LZ21A010002)the Fundamental Research Funds for the Central Universities.
文摘In this paper,we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space,where the random variables are not necessarily identically distributed.Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm.As an application,the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained.In the paper,it is also shown that if the sub-linear expectation space is rich enough,it will have no continuous capacity.The laws of the iterated logarithm are established without the assumption on the continuity of capacities.
基金supported by NSF of Shandong Province(Grant No.ZR2021MA018)National Key R&D Program of China(Grant No.2018YFA0703900)+1 种基金NSF of China(Grant No.11601281)the Young Scholars Program of Shandong University.
文摘A new Hartman-Wintner-type law of the iterated logarithm for independent random variables with mean-uncertainty under sublinear expectations is established by the martingale analogue of the Kolmogorov law of the iterated logarithm in classical probability theory.
基金supported by the National Natural Science Foundation of China(11361019)the Support Program of the Guangxi China Science Foundation(2015GXNSFAA139008)
文摘Consider a ρ-mixing sequence of identically distributed random variables with the underlying dis- tribution in the domain of attraction of the normal distribution. This paper proves that law of the iterated logarithm holds for ρ-mixing sequences of random variables. Our results generalize and improve Theorems 1.2-1.3 of Qi and Cheng （1996） from the i.i.d, case to ρ-mixing sequences.
基金Supported by grants from the NSF of China(Grant Nos.2024YFA101350,U23A2064 and 12031005)。
文摘We establish the Strassen's law of the iterated logarithm(LIL for short)for independent and identically distributed random variables with E[X_(1)]=ε[X_(1)]=0 and Cv[X_(1)^(2)]<∞ under a sublinear expectation space with a countably sub-additive capacity V.We also show the LIL for upper capacity with σ=σ under some certain conditions.
文摘With the help of large deviation of Brownian motion in the Hölder norm,local Strassen’s law of the iterated logarithm for increments of a Brownian motion in the Hölder norm is investigated.This paper promotes the corresponding results by Gantert and Wei.
基金Supported by National Natural Science Foundation of China （Grant No. 10871153）
文摘Let {β（s）,s ≥ O} be the standard Brownian motion in R^d with d ≥ 4 and let ｜Wr（t）｜ be the volume of the Wiener sausage associated with {β（s）, s ≥ O} observed until time t. Prom the central limit theorem of Wiener sausage, we know that when d ≥ 4 the limit distribution is normal. In this paper, we study the laws of the iterated logarithm for ｜Wr（t）｜ - e｜Wr（t）｜ in this case.
基金The author would like to thank the referees for comments on conditions （C1） and （C2）. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11171262） and the Specialized Research Fund for the Doctoral Program of Higher Education of China （Grant No. 20130141110076）.
文摘We consider laws of iterated random walks in random environments. logarithm for one-dimensional transient A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environments, including independent identically distributed environments and uniformly ergodic environments.
