Almost sure central limit theory for products of sums of partial sums

Consider a sequence of i.i.d.positive random variables.An universal result in almost sure limit theorem for products of sums of partial sums is established.We will show that the almost sure limit theorem holds under a fairly general condition on the weight dk= k-1 exp（lnβk）,0≤β〈1.And in a sense,our results have reached the optimal form.

SELF-NORMALIZED CENTRAL LIMIT THEOREM AND ESTIMATION OF VARIANCE OF PARTIAL SUMS FOR NEGATIVE DEPENDENT RANDOM VARIABLES

Let ｛X n,n≥1｝ be a stationary LNQD or NA sequence satisfying EX 1=μ,EX 2 1<∞ and (Var S n)/n→σ 2 as n→∞.In this paper a class of self-normalized central limit theorems and estimators of Var S n are studied.The weak and strong consistency of the estimators of Var S n are presented.

On Moments of the Maximum of Normed Partial Sums of ρ^--mixing Random Variables

In this paper, we obtain the moment conditions for the supermun of normed sums of ρ^--mixing random variables by using the Rosenthal-type inequality for Maximum partial sums of ρ^--mixing random variables. The result obtained generalize the results of Chen(2008) and extend those to negatively associated sequences and ρ^--mixing random variables.

ON UNIFORM APPROXIMATION BY PARTIAL SUMS OF JACOBI SERIES ON ELLIPTIC REGION

In this paper, we discuss the relation between the partial sums of Jacobi serier on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the partial sums of Jacobi series on an elliptic region.

Complete Convergence for Partial Sums of the Squares of ARCH Sequence

In this paper,we discuss the suffcient condition for complete convergence of partial sums of squares of the ARCH sequence,at the same time a moment inequality of maximum partial sums for squares of the ARCH sequence is proved.

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.

ON THE LIMITING BEHAVIOR OF THE MAXIMUM PARTIAL SUMS FOR ARRAYS OF ROWWISE NA RANDOM VARIABLES

Let {Xni, 1 ≤ n,i 〈 ∞} be an an array of rowwise NA random variables and {an, n ≥ 1} a sequence of constants with 0 〈 an ↑∞ . The limiting behavior of maximum partial sums 1/an max 1≤k≤n｜^k∑i=1 Xni｜ is investigated and some new results are obtained. The results extend and improve the corresponding theorems of rowwise independent random variable arrays by Hu and Taylor [1] and Hu and Chang [2].

On the Limiting Behavior of Weighted Partial Sums for B Valued Martingale Difference Sequences

Let {Xn, n≥1} be a martingale difference sequence and {a nk , 1?k?n,n?1} an array of constant real numbers. The limiting behavior of weighted partial sums ∑ k=1 n a nk X k is investigated and some new results are obtained.

Moments of the maximum of normed partial sums of ρ^--mixing random variables

Let {Xn,n ≥ 1} be a sequence of identically distributed ρ^--mixing random variables and set Sn =∑i^n=1 Xi,n ≥ 1,the suffcient and necessary conditions for the existence of moments of supn≥1 ｜Sn/n^1/r｜^p（0 〈 r 〈 2,p 〉 0） are given,which are the same as that in the independent case.

Studentized Increments of Partial Sums

Using suitable self-normalization for partial sums of i.i.d.random variables,Griffin and Kuelbs established the law of the iterated logarithm for all distributions in the domain of attraction of a normal law.We obtain the corresponding results for Studentized increments of partial sums under thesame condition.

Almost Sure Central Limit Theorem for Partial Sums of Markov Chain

Abstract The authors prove an almost sure central limit theorem for partial sums based on an irreducible and positive recurrent Markov chain using logarithmic means, which realizes the extension of the almost sure central limit theorem for partial sums from an i.i.d, sequence of random variables to a Markov chain.

An Improved Result in Almost Sure Central Limit Theory for Products of Partial Sums with Stable Distribution

Consider a sequence of i.i.d.positive random variables with the underlying distribution in the domain of attraction of a stable distribution with an exponent in (1,2].A universal result in the almost sure limit theorem for products of partial sums is established. Our results significantly generalize and improve those on the almost sure central limit theory previously obtained by Gonchigdanzan and Rempale and by Gonchigdanzan.In a sense,our results reach the optimal form.

Moment inequalities for the partial sums of random variables

This paper discusses the conditions under which Rosenthal type inequality is obtained from M-Z-B type inequality. And M-Z-B type inequality is proved for a wide class of random variables. Hence Rosenthal type inequalities for some classes of random variables are obtained.

Joint Limit Distributions of Exceedances Point Processes and Partial Sums of Gaussian Vector Sequence

In this paper, we study the joint limit distributions of point processes of exceedances and partial sums of multivariate Gaussian sequences and show that the point processes and partial sums are asymptotically independent under some mild conditions. As a result, for a sequence of standardized stationary Gaussian vectors, we obtain that the point process of exceedances formed by the sequence （centered at the sample mean） converges in distribution to a Poisson process and it is asymptotically independent of the partial sums. The asymptotic joint limit distributions of order statistics and partial sums are also investigated under different conditions.

Joint asymptotic distribution of exceedances point process and partial sum of stationary Gaussian sequence

Let {Xi}i=1^∞ be a standardized stationary Gaussian sequence with covariance function τ（n） =EX1Xn＋1, Sn =∑i=1^nXi,and X^-n=Sn/n.And let Nn be the point process formed by the exceedances of random level （x/√2 log n＋√2 log n-log（4π log n）/2√log n） √1-τ（n） ＋ X^-n by X1,X2,…, Xn. Under some mild conditions, Nn and Sn are asymptotically independent, and Nn converges weakly to a Poisson process on （0,1].

POSITIVITY AND BOUNDEDNESS OF TRIGONOMETRIC SUMS

We give a systematic account of results which assure positivity and boundedness of partial sums of cosine or sine series. New proofs of recent results are sketched.

On the Divergence of Fourier-Walsh Series

We prove that for any p perfect set of positive measure and for it＇s any density point x0 one can construct a measurable function f（x）, bounded on [0,1）, such that each measurable and bounded function, which coincides with f（x） on the set p has diverging Fourier-Walsh series on the point xo.

ON THE PERIODIC ORTHONORMAL WAVELET SYSTEM

This paper concerns with the point-wise convergence of a wavelet series for L-p functions and the characterization of Holder class by both the partial sums approximation and the best approximation.

Detecting Strength and Location of Jump Discontinuities in Numerical Data

In [1] and some following publications, Tadmor and Gelb took up a well known property of conjugate Fourier series in 1-d, namely the property to detect jump discontinuities in given spectral data. In fact, this property of conjugate series is known for quite a long time. The research in papers around the year 1910 shows that there were also other means of detecting jumps observed and analysed. We review the classical results as well as the results of Gelb and Tadmor and demonstrate their discrete case using different estimates in all detail. It is worth noting that the techniques presented are not global but local techniques. Edges are a local phenomenon and can only be found appropriately by local means. Furthermore, applying a different approach in the proof of the main estimate leads to weaker preconditions in the discrete case. Finally an outlook to a two-dimensional approach based on the work of Móricz, in which jumps in the mixed second derivative of a 2-d function are detected, is made.

ALMOST EVERYWHERE CONVERGENCE AND APPROXIMATION OF SPHERICAL HARMONIC EXPANSIONS

This paper deals with the order of magnitude of the partial sums of the spherical harmonic series and its convergence rate in Bessel potential spaces. The partial results obtained in the paper are the analogue of those on the circle.