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
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 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
Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 ＋... ＋ Xn, n≥1. The author proves that, if EX^2I{｜X｜≥t} = 0（（log log t）^-1） as t→∞, then for ...Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 ＋... ＋ Xn, n≥1. The author proves that, if EX^2I{｜X｜≥t} = 0（（log log t）^-1） as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1＋a（1/√1＋a-ε）b＋1 ∑n=1^∞（logn）^a（loglogn）^b/nP{max κ≤n｜Sκ｜≤√σ^2π^2n/8loglogn（ε＋an）}=4/π（1/2（1＋a）^3/2）^b＋1 Г（b＋1）,whenever an = o（1/log log n）. The author obtains the sufficient and necessary conditions for this kind of results to hold.展开更多
For a set of i.i.d.r.v. indexed by positive integer d-dimensional lattice points, and for some general normalizing sequence, we determine necessary and sufficient conditions for the law of iterated logarithm. As its a...For a set of i.i.d.r.v. indexed by positive integer d-dimensional lattice points, and for some general normalizing sequence, we determine necessary and sufficient conditions for the law of iterated logarithm. As its application, we give conditions for the existence of moments of the supremum of normed partial sums.展开更多
We prove a new Donsker’s invariance principle for independent and identically distributed random variables under the sub-linear expectation.As applications,the small deviations and Chung’s law of the iterated logari...We prove a new Donsker’s invariance principle for independent and identically distributed random variables under the sub-linear expectation.As applications,the small deviations and Chung’s law of the iterated logarithm are obtained.展开更多
In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables w...In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.展开更多
Consider a sequence of negatively associated and identically distributed random variableswith the underlying distribution in the domain of attraction of a stable distribution with an exponentin(0,2).A Chover's law...Consider a sequence of negatively associated and identically distributed random variableswith the underlying distribution in the domain of attraction of a stable distribution with an exponentin(0,2).A Chover's law of the iterated logarithm is established for negatively associated randomvariables.Our results generalize and improve those on Chover's law of the iterated logarithm(LIL)type behavior previously obtained by Mikosch(1984),Vasudeva(1984),and Qi and Cheng(1996)fromthe i.i.d,case to NA sequences.展开更多
Negatively associated sequences have been studied extensively in recent years, Asymptotically negative association is a generalization of negative association, In this paper a Berry Esseen theorem and a law of the ite...Negatively associated sequences have been studied extensively in recent years, Asymptotically negative association is a generalization of negative association, In this paper a Berry Esseen theorem and a law of the iterated logarithm are obtained for asymptotically negatively associated sequences.展开更多
Let {X, X1, X2,...} be a strictly stationaryφ-mixing sequence which satisfies EX = 0,EX^2（log2{X}）^2〈∞and φ（n）=O（1/log n）^Tfor some T〉2.Let Sn=∑k=1^nXk and an=O（√n/（log2n）^γ for some γ〉1/2.We prove ...Let {X, X1, X2,...} be a strictly stationaryφ-mixing sequence which satisfies EX = 0,EX^2（log2{X}）^2〈∞and φ（n）=O（1/log n）^Tfor some T〉2.Let Sn=∑k=1^nXk and an=O（√n/（log2n）^γ for some γ〉1/2.We prove that limε→√2√ε^2-2∑n=3^∞1/nP（｜Sn｜≥ε√ESn^2log2n＋an）=√2.The results of Gut and Spataru （2000） are special cases of ours.展开更多
Let XH = {xH（t）,t ∈ R＋} be a subfractional Brownian motion in Rd. We provide asufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that XH has the property of s...Let XH = {xH（t）,t ∈ R＋} be a subfractional Brownian motion in Rd. We provide asufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that XH has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of XH, we establish Chung＇s law of the iterated logarithm for XH.展开更多
For right censored data, the law of the iterated logarithm of the Kaplan-Meier integral is established. As an application, the authors prove the law of the iterated logarithm for weighted least square estimates of ran...For right censored data, the law of the iterated logarithm of the Kaplan-Meier integral is established. As an application, the authors prove the law of the iterated logarithm for weighted least square estimates of randomly censored linear regression model.展开更多
A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish a Chung-type law of the iterated logarithm for continuous time...A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish a Chung-type law of the iterated logarithm for continuous time random walk with jumps and waiting times in the domains of attraction of stable laws.展开更多
For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of ...For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent;（ii） For all constants λ ∈ [0, ∞）,lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.展开更多
By estimating small ball probabilities for l^P-valued Gaussian processes, a Chung-type law of the iterated logarithm of l^P-valued Gaussian processes is given.
By using the Ito's calculus, a law of the iterated logarithm is established for the processes with independent increments (PⅡ). Let X = {Xt, t ≥ 0} be a PII with Ext=0,V(t)=Ext2<∞and limt∞V(t)=∞ If one of ...By using the Ito's calculus, a law of the iterated logarithm is established for the processes with independent increments (PⅡ). Let X = {Xt, t ≥ 0} be a PII with Ext=0,V(t)=Ext2<∞and limt∞V(t)=∞ If one of the following conditions is satisfied,(2) Suppose the Levy's measure of X may be written as v(dt,ds) = Ft(dx) dV(t) and there is a σ-finite measure G such tnat ,展开更多
Let X={X(t),t 0} be a process with independent increments (PII)such that E=0, D X(t)E 2<∞, lim t→∞D X(t)t=1, and there exists a majoring measure G for the jump △X of X . Under these assu...Let X={X(t),t 0} be a process with independent increments (PII)such that E=0, D X(t)E 2<∞, lim t→∞D X(t)t=1, and there exists a majoring measure G for the jump △X of X . Under these assumptions, using rather a direct method, a Strassen's law of the iterated logarithm (Strassen LIL) is established. As some special cases,the Strassen LIL for homogeneous PII and for partial sum process of i.i.d.random variables are comprised.展开更多
In this note,we establish a compact law of the iterated logarithm under the upper capacity for independent and identically distributed random variables in a sub-linear expectation space.For showing the result,a self-n...In this note,we establish a compact law of the iterated logarithm under the upper capacity for independent and identically distributed random variables in a sub-linear expectation space.For showing the result,a self-normalized law of the iterated logarithm is established.展开更多
In this paper,we investigate functional limit problem for path of a Brownian sheet,Chung’s functional law of the iterated logarithm for a Brownian sheet is obtained.The main tool in the proof is large deviation and s...In this paper,we investigate functional limit problem for path of a Brownian sheet,Chung’s functional law of the iterated logarithm for a Brownian sheet is obtained.The main tool in the proof is large deviation and small deviation for a Brownian sheet.展开更多
Consider tile partial linear model Y=Xβ+ g(T) + e. Wilers Y is at risk of being censored from the right, g is an unknown smoothing function on [0,1], β is a 1-dimensional parameter to be estimated and e is an unobse...Consider tile partial linear model Y=Xβ+ g(T) + e. Wilers Y is at risk of being censored from the right, g is an unknown smoothing function on [0,1], β is a 1-dimensional parameter to be estimated and e is an unobserved error. In Ref[1,2], it wes proved that the estimator for the asymptotic variance of βn(βn) is consistent. In this paper, we establish the limit distribution and the law of the iterated logarithm for,En, and obtain the convergest rates for En and the strong uniform convergent rates for gn(gn).展开更多
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National Natural Science Foundation of China (No.10471126)
文摘Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 ＋... ＋ Xn, n≥1. The author proves that, if EX^2I{｜X｜≥t} = 0（（log log t）^-1） as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1＋a（1/√1＋a-ε）b＋1 ∑n=1^∞（logn）^a（loglogn）^b/nP{max κ≤n｜Sκ｜≤√σ^2π^2n/8loglogn（ε＋an）}=4/π（1/2（1＋a）^3/2）^b＋1 Г（b＋1）,whenever an = o（1/log log n）. The author obtains the sufficient and necessary conditions for this kind of results to hold.
基金the National Natural Science Foundation of China(10271120)
文摘For a set of i.i.d.r.v. indexed by positive integer d-dimensional lattice points, and for some general normalizing sequence, we determine necessary and sufficient conditions for the law of iterated logarithm. As its application, we give conditions for the existence of moments of the supremum of normed partial sums.
基金This research supported by Grants from the National Natural Science Foundation of China(No.11225104)and the Fundamental Research Funds for the Central Universities.
文摘We prove a new Donsker’s invariance principle for independent and identically distributed random variables under the sub-linear expectation.As applications,the small deviations and Chung’s law of the iterated logarithm are obtained.
基金Research supported by National Natural Science Foundation of China
文摘In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.
基金supported by the National Natural Science Foundation of China under Grant No.10661006the Support Program of the New Century Guangxi China Ten-Hundred-Thousand Talents Project under Grant No.2005214the Guangxi, China Science Foundation under Grant No.2010GXNSFA013120
文摘Consider a sequence of negatively associated and identically distributed random variableswith the underlying distribution in the domain of attraction of a stable distribution with an exponentin(0,2).A Chover's law of the iterated logarithm is established for negatively associated randomvariables.Our results generalize and improve those on Chover's law of the iterated logarithm(LIL)type behavior previously obtained by Mikosch(1984),Vasudeva(1984),and Qi and Cheng(1996)fromthe i.i.d,case to NA sequences.
基金National Natural Science Foundation of China(No.10471126)
文摘Negatively associated sequences have been studied extensively in recent years, Asymptotically negative association is a generalization of negative association, In this paper a Berry Esseen theorem and a law of the iterated logarithm are obtained for asymptotically negatively associated sequences.
基金National Natural Science Foundation of China (No.10571159)
文摘Let {X, X1, X2,...} be a strictly stationaryφ-mixing sequence which satisfies EX = 0,EX^2（log2{X}）^2〈∞and φ（n）=O（1/log n）^Tfor some T〉2.Let Sn=∑k=1^nXk and an=O（√n/（log2n）^γ for some γ〉1/2.We prove that limε→√2√ε^2-2∑n=3^∞1/nP（｜Sn｜≥ε√ESn^2log2n＋an）=√2.The results of Gut and Spataru （2000） are special cases of ours.
基金Supported by NSFC(Grant Nos.11201068,11671041)“the Fundamental Research Funds for the Central Universities”in UIBE(Grant No.14YQ07)
文摘Let XH = {xH（t）,t ∈ R＋} be a subfractional Brownian motion in Rd. We provide asufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that XH has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of XH, we establish Chung＇s law of the iterated logarithm for XH.
基金Project supported by the National Natural Science Foundation of China (No.10231030) the Research Fund for the Doctoral Program of Higher Education.
文摘For right censored data, the law of the iterated logarithm of the Kaplan-Meier integral is established. As an application, the authors prove the law of the iterated logarithm for weighted least square estimates of randomly censored linear regression model.
基金Supported in part by the National Natural Science Foundation of China under Grant No.11671115 the Natural Science Foundation of Zhejiang Province under Grant No.LY14A010025
文摘A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish a Chung-type law of the iterated logarithm for continuous time random walk with jumps and waiting times in the domains of attraction of stable laws.
基金the Natural Sciences and Engineering Research Council of Canada
文摘For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent;（ii） For all constants λ ∈ [0, ∞）,lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.
基金Research supported by NSFC (10401037)supported by SRFDP (2002335090) China Postdoctoral Science Foundation
文摘By estimating small ball probabilities for l^P-valued Gaussian processes, a Chung-type law of the iterated logarithm of l^P-valued Gaussian processes is given.
文摘By using the Ito's calculus, a law of the iterated logarithm is established for the processes with independent increments (PⅡ). Let X = {Xt, t ≥ 0} be a PII with Ext=0,V(t)=Ext2<∞and limt∞V(t)=∞ If one of the following conditions is satisfied,(2) Suppose the Levy's measure of X may be written as v(dt,ds) = Ft(dx) dV(t) and there is a σ-finite measure G such tnat ,
文摘Let X={X(t),t 0} be a process with independent increments (PII)such that E=0, D X(t)E 2<∞, lim t→∞D X(t)t=1, and there exists a majoring measure G for the jump △X of X . Under these assumptions, using rather a direct method, a Strassen's law of the iterated logarithm (Strassen LIL) is established. As some special cases,the Strassen LIL for homogeneous PII and for partial sum process of i.i.d.random variables are comprised.
文摘In this note,we establish a compact law of the iterated logarithm under the upper capacity for independent and identically distributed random variables in a sub-linear expectation space.For showing the result,a self-normalized law of the iterated logarithm is established.
基金supported by the Natural Science Foundation of Guangxi(Grant No.2020GXNSFAA159118)Guangxi Science and Technology Project(Grant No.Guike AD20297006)the Innovation Project of School of Mathematics and Computing Science of GUET Graduate Education(Nos.2021YJSCX05,2022YJSCX04)。
文摘In this paper,we investigate functional limit problem for path of a Brownian sheet,Chung’s functional law of the iterated logarithm for a Brownian sheet is obtained.The main tool in the proof is large deviation and small deviation for a Brownian sheet.
文摘Consider tile partial linear model Y=Xβ+ g(T) + e. Wilers Y is at risk of being censored from the right, g is an unknown smoothing function on [0,1], β is a 1-dimensional parameter to be estimated and e is an unobserved error. In Ref[1,2], it wes proved that the estimator for the asymptotic variance of βn(βn) is consistent. In this paper, we establish the limit distribution and the law of the iterated logarithm for,En, and obtain the convergest rates for En and the strong uniform convergent rates for gn(gn).
