In this paper, we define the generalized linear models (GLM) based on the observed data with incomplete information and random censorship under the case that the regressors are stochastic. Under the given conditions, ...In this paper, we define the generalized linear models (GLM) based on the observed data with incomplete information and random censorship under the case that the regressors are stochastic. Under the given conditions, we obtain a law of iterated logarithm and a Chung type law of iterated logarithm for the maximum likelihood estimator (MLE) in the present model.展开更多
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 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
Consider the positive d-dimensional lattice Z^d（d≥2） with partial ordering ≤, let {XK; K∈Z＋^d} be i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with mean zero and ...Consider the positive d-dimensional lattice Z^d（d≥2） with partial ordering ≤, let {XK; K∈Z＋^d} be i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with mean zero and covariance operator ∑ and set partial sums SN =∑K≤nXK,K,N∈Z＋^d. Under some moment conditions, we obtain the precise asymptotics of a kind of weighted infinite series for partial sums SN as ε↓ by using the truncation and approximation methods. The results are related to the convergence rates of the law of the logarithm in Hilbert space, and they also extend the results of （Gut and Spataru, 2003）.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 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
The vertical profiles of longshore currents have been examined experimentally over plane and barred beaches. In most cases, the vertical profiles of longshore currents are expressed by the logarithmic law. The power l...The vertical profiles of longshore currents have been examined experimentally over plane and barred beaches. In most cases, the vertical profiles of longshore currents are expressed by the logarithmic law. The power law is not commonly used to describe the profile of longshore currents. In this paper, however, a power-type formula is proposed to describe the vertical profiles of longshore currents. The formula has two parameters: the power law index (a) and the depth-averaged velocity. Based on previous studies, power law indices were set as a=1/10 and a=1/7. Depth-averaged velocity can be obtained through measurement. The fitting of the measured velocity profiles to a=1/10 and a=1/7 was assessed for the vertical longshore profiles. The vertical profile of longshore currents is well described by the power-type formula with a=1/10 for a plane beach. However, for a barred beach, different values of a needed to be used for different regions. For the region from the bar trough to the offshore side of the bar crest, the vertical profiles of longshore currents given by the power-type formula with a=1/10 and a=1/7 fit the data well. However, the fit was slightly better with a=1/10 than that with a=1/7. For the data over the trough region of cross-shore distribution of the depth-averaged longshore currents, the power formula with a=1/3 provided a good fit. The formulas with a=1/10 and a=1/7 were further examined using published data from four sources covering laboratory and field experiments. The results indicate that the power-type formula fits the data well for the laboratory and field data with a=1/10.展开更多
We investigate the strong law of large numbers(SLLN)for a large class of mean based on the Extended Negatively Dependent(END)sequences.The sufficient conditions are obtained for the mean of SLLN in this paper.As an im...We investigate the strong law of large numbers(SLLN)for a large class of mean based on the Extended Negatively Dependent(END)sequences.The sufficient conditions are obtained for the mean of SLLN in this paper.As an important application,the SLLN of Marcinkiewicz mean and logarithmic mean are presented immediately.In addition,we do some simulations for the mean of SLLN based on END random variables.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
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
文摘In this paper, we define the generalized linear models (GLM) based on the observed data with incomplete information and random censorship under the case that the regressors are stochastic. Under the given conditions, we obtain a law of iterated logarithm and a Chung type law of iterated logarithm for the maximum likelihood estimator (MLE) in the present model.
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 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）.
基金Project (No. 10471126) supported by the National Natural Science Foundation of China
文摘Consider the positive d-dimensional lattice Z^d（d≥2） with partial ordering ≤, let {XK; K∈Z＋^d} be i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with mean zero and covariance operator ∑ and set partial sums SN =∑K≤nXK,K,N∈Z＋^d. Under some moment conditions, we obtain the precise asymptotics of a kind of weighted infinite series for partial sums SN as ε↓ by using the truncation and approximation methods. The results are related to the convergence rates of the law of the logarithm in Hilbert space, and they also extend the results of （Gut and Spataru, 2003）.
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.
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.
基金supported by Science and Technology Support Program of Fujian Province,China(Grant No.2015Y0035)the National Natural Science Foundation of China(Grant No.10672034)
文摘The vertical profiles of longshore currents have been examined experimentally over plane and barred beaches. In most cases, the vertical profiles of longshore currents are expressed by the logarithmic law. The power law is not commonly used to describe the profile of longshore currents. In this paper, however, a power-type formula is proposed to describe the vertical profiles of longshore currents. The formula has two parameters: the power law index (a) and the depth-averaged velocity. Based on previous studies, power law indices were set as a=1/10 and a=1/7. Depth-averaged velocity can be obtained through measurement. The fitting of the measured velocity profiles to a=1/10 and a=1/7 was assessed for the vertical longshore profiles. The vertical profile of longshore currents is well described by the power-type formula with a=1/10 for a plane beach. However, for a barred beach, different values of a needed to be used for different regions. For the region from the bar trough to the offshore side of the bar crest, the vertical profiles of longshore currents given by the power-type formula with a=1/10 and a=1/7 fit the data well. However, the fit was slightly better with a=1/10 than that with a=1/7. For the data over the trough region of cross-shore distribution of the depth-averaged longshore currents, the power formula with a=1/3 provided a good fit. The formulas with a=1/10 and a=1/7 were further examined using published data from four sources covering laboratory and field experiments. The results indicate that the power-type formula fits the data well for the laboratory and field data with a=1/10.
基金Supported by the National Natural Science Foundation of China(Grant No.11701004)the Natural Science Foundation of Anhui Province(Grant Nos.1808085QA03,1808085QF212,1808085QA17)+2 种基金Provincial Natural Science Research Project of Anhui Colleges(Grant Nos.KJ2016A027,KJ2017A027,KJ2019A0006)Excellent Young Talents Research Project of Anhui Colleges(Grant No.gxyq2018102)Key Research Project of Suzhou University(Grant No.2017yzd16)。
文摘We investigate the strong law of large numbers(SLLN)for a large class of mean based on the Extended Negatively Dependent(END)sequences.The sufficient conditions are obtained for the mean of SLLN in this paper.As an important application,the SLLN of Marcinkiewicz mean and logarithmic mean are presented immediately.In addition,we do some simulations for the mean of SLLN based on END random variables.
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 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.
