A Note on the Almost Sure Central Limit Theorem for Partial Sums of ρ^(−)-Mixing Sequences

Let be a strictly stationary sequence of ρ?-mixing random variables. We proved the almost sure central limit theorem, containing the general weight sequences, for the partial sums , where , . The result generalizes and improves the previous results.

Complete Convergence Properties of the Sums for -mixing Sequences of Random Variables

This paper discusses complete convergence properties of the sums of -mixing random sequences.As a result,we improve the corresponding results of Wu Qunying(2001). And extended the Baum and Katz complete convergence to the case of -mixing random sequences by moment inequality and truncating without necessarily adding any extra conditions.

Strong Laws of Large Numbers for Weighted Sums of Ч-mixing Sequence

In this paper, strong laws of large numbers for weighted sums of ■-mixing sequence are investigated. Our results extend the corresponding results for negatively associated sequence to the case of ■-mixing sequence.

Strong Law of Large Numbers and Complete Convergence for Sequences of -Mixing Random Variables

We give some theorems of strong law of large numbers and complete convergence for sequences of φ-mixing random variables. In particular, Wittmann＇s strong law of large numbers and Teicher＇s strong law of large nnumbers for independent random variables are generalized to the case of φ -minxing random variables.

The Law of the Iterated Logarithm for the Sums of φ-Mixing Sequences with Duple Suffixes

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.

Complete Convergence for Weighted Sums of φ-mixing Sequence

In this paper, the complete convergence and strong law of large numbers for weighted sums of φ-mixing sequence with different distribution are investigated under some weaker moment conditions. Our results extend ones of independent sequence with identical distribution to the case of φ-mixing sequence with different distribution.

Complete Convergence and Weak Law of Large Numbers for <i><span style="text-decoration:overline;">ρ</span></i>-Mixing Sequences of Random Variables

In this paper, the complete convergence and weak law of large numbers are established for ρ-mixing sequences of random variables. Our results extend and improve the Baum and Katz complete convergence theorem and the classical weak law of large numbers, etc. from independent sequences of random variables to ρ-mixing sequences of random variables without necessarily adding any extra conditions.

STUDY ON THE SEQUENCE STRUCTURE OF BUTADIENE-STYRENE RUBBER BY ^(13)C-NMR METHOD V.CALCULATION METHOD OF DIAD COMPOSITION

The calculating equations which could be used to obtain diad composition from the peakintensities of carbon spectra were derived. The result of the microstructure calculated from thediad compositions was in good agreement with that obtained through IR or;H-NMR measurement.

STUDY ON THE SEQUENCE STRUCTURE OF SBR BY ^(13)C—NMR METHOD Ⅰ. ASSIGNMENT FOR UNSATURAT CARBONS SPECTRA

The sequence structures of emulsion-processed SBR and solution-processed (by lithium catalyst) SBR were investigated by ^(13)C-NMR spectroscopy. Seventeen peaks within unsaturated carbon region were recorded under the adopted experimental conditions. Assignments for these peaks were made by empirical-parameter-evaluation method.

LARGE DEVIATIONS FOR SOME DEPENDENT SEQUENCES

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.

Some Convergence Results for Arbitrary Sequences under Moment Condition

Let {X n , n ≥ 1} be an arbitrary sequence of random variables. Some convergence results for the partial sums of arbitrary sequence of random variables are obtained, which generalize the known results for independent sequences, NA sequences, ρ-mixing sequences and φ-mixing sequences, and so on.

Convergence Rate of Estimator forNonparametric Regression Model under ρ-mixing Errors

In this paper, we investigate the nonparametric regression model based on ρ-mixing errors, which are stochastically dominated by a nonnegative random variable. Weobtain the convergence rate for the weighted estimator of unknown function g（x） in pth-mean, which yields the convergence rate in probability. Moreover, an example of the nearestneighbor estimator is also illustrated and the convergence rates of estimator are presented.

SUBSTITUENT CHEMICAL SHIFT (SCS) AND THE SEQUENCE STRUCTURE OF ETHYLENE-VINYL ALCOHOL COPOLYMERS

Three ethylene-vinyl alcohol copolymers were studied by means of the substituent chemical shift (SCS) method. The SCS parameters of hydroxy (-OH) in two different solvents were obtained: in deuterium oxide/phenol (20/80 W/W ) the parameters are S_1 = 42.77±0.08ppm, S_2 = 7.15±0.06 ppm, S_3 (s)= -4.08±0.02ppm, S_3 (t) =-3.09±0.20ppm,S_4 = 0.48±0.03ppm, S_5 = 0.26±0.05ppm. In o-dichloro-benzen-d_4 S_1(s)=44.79±0.61ppm, S_2=7.40±0.00ppm, S_3(s)=-4.51±0.17ppm, S_3(t)=-3.13 ±0.00 ppm, S_4 =0.63±0.04ppm, S_5=0.36±0.00ppm.Simultaneously the ^(13)CNMR spectra of EVA copolymers were assigned by using the SCS parameters obtained.

A LOW OF THE ITERATED LOGARITHM FOR DOUBLE ARRAY SUM OF φp-MIXING SEQUENCE

I. INTRODUCTION AND SOME LEMMASLet{εi} be independent random variables, E（εi） = 0, E（εi2） =σ2, sup E[εi|r【∞ for some r】2, {ani} be a double array of constants satisfying proper conditions. Ref. [1] gave

Rates of uniform convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions

Assume that {X n } is a strictly stationary β-mixing random sequence with the β-mixing coefficient β k = O(k ?r ), 0 < r ? 1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in L p, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((n r/(1+r)/log n)?1/2) or O((n r/(1+r)/log n)?3/4) and that the Central Limit Theorem does not always hold for the empirical processes.

Complete convergence of ρ-mixing sequence

Let {Xn, n≥1} be a sequence of random variables and let Sn=∑1≤i≤nXi,n-=σ（Xi1≤i≤n）,n+=σ（Xi,i≥n）,n≥1.

LARGE DEVIATIONS FOR STATIONARY Φ-MIXING SEQUENCES IN τ-TOPOLOGY

The results of Bryc on large deviations for empirical measures of stationary Φ-mixing sequences are extended. Bryc states his results in the usual weak topology on the space ofprobability measures. In this paper, under somewhat weaker assumptions than those of Bryc,the author extends Bryc's results by taking the finer topology which is generated by the integralsover bounded measurable functions.

^(13)C-NMR STUDY OF ETHYLENE-1-OCTENE COPOLYMERS

A ^(13)C-NMR method is presented for a quantitative determination of the respective monomercomposition and sequence distributions in ethylene-1-octene copolymers prepared with supportedtitanium/magnesium catalyst. On the basis of the sequence distributions, the ethylene-1-octenecopolymerization mechanism was studied. It was found that the observed sequence distributions inethylene-1-octene copolymers are satisfactorily predicted by the two-site model M/M,in whichthe copolymerization proceeds according to first-order Markovian statistics at the two differentsites.

SEQUENCE STRUCTURE OF ETHYLENE-VINYL SUBSTITUTED COPOLYMER WITH ^(13)C NMR IV——CHLORINATED POLYETHYLENE

The substituent chemical shift (SCS) has been applied to the assignment of the ^(13)C NMRspectrum of chlorinated polyethylene (CPE). CPE of different chlorine contents has been em-ployed and their sequence structure discussed. The results show that characteristic of CPEwith medium chlorine content is the dichloroethane structure in molecular chain. SCS param-etets have been obtained from the ^(13)C NMR spectra. It was found that the effects of chlorinecontent and temperature on SCS are negligible, but the substituent parameter S_1 reduced by0.39 ppm when C_2Cl_4 was added to solvent ODCB.

Strong Convergence of Partitioning Estimation for Nonparametric Regression Function under Dependence Samples

In this paper, we study the strong consistency for partitioning estimation of regression function under samples that axe φ-mixing sequences with identically distribution.Key words: nonparametric regression function; partitioning estimation; strong convergence;φ-mixing sequences.