LARGE DEVIATION FOR THE EMPIRICAL CORRELATION COEFFICIENT OF TWO GAUSSIAN RANDOM VARIABLES

In this article,the author obtains the large deviation principles for the empir- ical 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.     

EQUIVALENCE OF COMPLETE CONVERGENCE AND LAW OF LARGE NUMBERS FOR B-VALUED RANDOM ELEMENTS＇

Under some conditions on probability, this note discusses the equivalence between the complete convergence and the law of large number forB- valued independent random elements. The results of [10] become a simple corollary of the results here. At the same time, the author uses them to investigate the equivalence of strong and weak law of large numbers, and there exists an example to show that the conditions on probability are weaker. Project supported by the National Natural Science Foundation of China.     

SMALL PERTURBATION CRAMER METHODS AND MODERATE DEVIATIONS FOR MARKOV PROCESSES

ＳＭＡＬＬＰＥＲＴＵＲＢＡＴＩＯＮＣＲＡＭＥＲＭＥＴＨＯＤＳＡＮＤＭＯＤＥＲＡＴＥＤＥＶＩＡＴＩＯＮＳＦＯＲＭＡＲＫＯＶＰＲＯＣＥＳＳＥＳＧａｏＦｕｑｉｎｇ（高付清）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｍａｔｉｃｓ，ＨｕｂｅｉＵａｉｖｅｒｓｉｔｙ，Ｗｕ...     

THE ERDOS-RENYI LAWS OF LARGE NUMBERS FOR NON-IDENTICALLY DISTRIBUTED RANDOM VARIABLES

The Erdos-Renyi law of large numbers (1970) is the first important result forasymptotic behaviours of increments of partial sams of a sequence of random variableswith apan [ClogN]. Some generalizations have been done sinoe then, such as conver-gence rate of the limit, some results when order of span being either higher or lowerthan log N. But all these results are only obtained in the case of i. i. d. random variables.This paper aims at the generalization of these results to the ease when random variablesare independent, but not necessarily identically distributed. To this end Chernoff Theoremis generalized to the corlesponding case at first.     

DIRECT EXPANSIONS FOR THE DISTRIBUTION FUNCTIONS AND DENSITY FUNCTIONS OF χ2-TYPE AND t-TYPE DISTRIBUTED RANDOM VARIABLES

Suppose that Z1,Z2…,Zn are independent normal random variables with common mean μ and variance σ^2. Then S^2=∑n n=1 (zi-z)^2/σ^2 and T =（n-1的平方根）-Z/（S^2/n的平方根） have x2n-1 distribution and tn-1 distribution respectively. If the normal assumption fails, there will be the remainders of the distribution functions and density functions. This paper gives the direct expansions of distribution functions and density functions of S^2 and T up to o(n^-1). They are more intuitive and convenient than usual Edgeworth expansions.     

LARGE DEVIATIONS FOR INFINITE DIMENSIONAL AND REVERSIBLE REACTION－DIFFUSION PROCESSES

ＬＡＲＧＥＤＥＶＩＡＴＩＯＮＳＦＯＲＩＮＦＩＮＩＴＥＤＩＭＥＮＳＩＯＮＡＬＡＮＤＲＥＶＥＲＳＩＢＬＥＲＥＡＣＴＩＯＮ－ＤＩＦＦＵＳＩＯＮＰＲＯＣＥＳＳＥＳＣＨＥＮＪＩＮＷＥＮ（陈金文）（ＤｅｐｅｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，Ｔ...     

LARGE DEVIATIONS FOR FIELDS WITH STATIONARY INDEPENDENT INCREMENTS

ＬＡＲＧＥＤＥＶＩＡＴＩＯＮＳＦＯＲＦＩＥＬＤＳＷＩＴＨＳＴＡＴＩＯＮＡＲＹＩＮＤＥＰＥＮＤＥＮＴＩＮＣＲＥＭＥＮＴＳＧＡＯＦＵＱＩＮＧ（高付清）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｕｂｅｉＵｎｉｖｅｒｓｉｔｙ，Ｗｕｈａｎ４３００７２...     

STRONG LIMIT THEOREMS FOR BLOCKWISE m-DEPENDENT RANDOM VARIABLES AND A GENERALIZATION OF THE CONJECTURES OF MORICZ

The authors introduce the definition of block-wise m-dependent with respect to somepositive real sequence for random variables taking values in a separable Banach space,prove some general results on the strong convergence of blockwise m-dependent randomvariables,and in particular,give positive answers to Móricz conjectures([5])by asharper theorem.     

DIRECT EXPANSIONS FOR THE DISTRIBUTIONFUNCTIONS AND DENSITY FUNCTIONS OF χ~2-TYPEAND t-TYPE DISTRIBUTED RANDOM VARIABLES

61.IntroductionSometimesthenormalassumptionismadeintheproblemsoftheestimationsandtest-inghypotheses,thatis,wesupposetheindependentobservationsZ1,Z2,'',Znarenormalrandomvariableswithmeanpandvariancea2(pandu2areknownorunknownparam-eters).Forinstance,wewallttoseeifadruginducessleep,atreatmentandcontrolarecompared.Wemight,foreachgroupofnrandonilyselectedpatients,recordsleepingtimewithoutthedrug(oradministrationQfaplacebo)andthenaftersometimeadndnisterthedrugandrecordsleepingtimeagain.LetZibethe…     

ON THE STRONG LAW OF LARGE NUMBERS

This paper introduces the concept of BC sequences and investigates some conditions which imply the strong law of large numbers for these sequences. The authors also study the strong law of large numbers for general random variable sequences. As applications of the result the authors characterize p-smoothableness of Banach space. Some generalizations of Petrov theorem, the Marcinkiewicz-Zygmund theorem and Hoffmann-Jorgensen and Pisier theorem are obtained.     

次线性期望下独立随机变量列的大偏差和中偏差

本文研究在次线性期望下的独立随机变量列的大偏差和中偏差原理. 利用次可加方法, 我们得 到次线性期望下的大偏差原理. 与次线性期望下的中心极限定理相应的中偏差原理也被建立.     

The strong law of large numbers for pairwise negatively dependent random variables

Necessary and sufficient conditions for the validity of the strong law of large numbers for pairwise negatively dependent random variables with infinite means are formulated.     

Large deviations for product of sums of random variables

In this paper, we obtain sample path and scalar large deviation principles for the product of sums of positive random variables. We study the case when the positive random variables are independent and identically distributed and bounded away from zero or the left tail decays to zero sufficiently fast. The explicit formula for the rate function of a scalar large deviation principle is given in the case when random variables are exponentially distributed.     

Exponential inequalities for associated random variables and strong laws of large numbers

Some exponential inequalities for partial sums of associated random variables are established. These inequalities improve the corresponding results obtained by Ioannides and Roussas (1999), and Oliveira (2005). As application, some strong laws of large numbers are given. For the case of geometrically decreasing covariances, we obtain the rate of convergence n-1/2(log log n)1/2(logn) which is close to the optimal achievable convergence rate for independent random variables under an iterated logarithm, while Ioannides and Roussas (1999), and Oliveira (2005) only got n-1/3(logn)2/3 and n-1/3(logn)5/3, separately.     

A strong law of large numbers for fuzzy random variables

A limit of a sequence of fuzzy numbers is defined and its some properties are shown. Based on these concept and properties, an independent sequence of fuzzy random variables is considered and a strong law of large numbers for fuzzy random variables is shown.     

Moderate deviations principle for products of sums of random variables

Let(Xn)n≥1 be a sequence of independent identically distributed(i.i.d.) positive random variables with EX1 = μ,Var(X1) = σ2.In the present paper,we establish the moderate deviations principle for the products of partial sums(Πnk=1Sk/n!μn)1/(γbn√(2n))1where γ = σ/μ denotes the coefficient of variation and(bn) is the moderate deviations scale.     

Large deviations and moderate deviations for kernel density estimators of directional data

Let fn be the non-parametric kernel density estimator of directional data based on a kernel function K and a sequence of independent and identically distributed random variables taking values in d-dimensional unit sphere Sd-1. It is proved that if the kernel function is a function with bounded variation and the density function f of the random variables is continuous, then large deviation principle and moderate deviation principle for {sup x∈sd-1 ｜fn（x） - E（fn（x））｜, n ≥ 1} hold.     

Laws of large numbers of negatively correlated random variables for capacities

Our aim is to present some limit theorems for capacities.We consider a sequence of pairwise negatively correlated random variables.We obtain laws of large numbers for upper probabilities and 2-alternating capacities,using some results in the classical probability theory and a non-additive version of Chebyshev’s inequality and Boral-Contelli lemma for capacities.     

Kolmogorov-Feller型弱大数律的一注记

In this paper, the Kolmogorov-Feller type weak law of large numbers are obtained,which extend and improve the related known works in the literature.