Central limit theorem and almost sure central limit theorem for the product of some partial sums

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.     

A central limit theorem for random sums of random variables

Anscombe (1952) (also see Chung (1974)) has developed a central limit theoremof random sums of independent and identically distributed random variables. Applicability of this theorem in practice, however, is limited since the normalization requires random factors. In this paper we establish sufficient conditions under which the central limit theorem holds when such random factors are replaced by the underlying asymptotic mean and standard ddeviation. An application of this result in the context of shock models is also given.     

A functional central limit theorem for a generalized occupancy problem

A functional central limit theorem is proved for a class of finitely exchangeable random variables which are based on an occupancy scheme.     

On the almost sure central limit theorem for randomly indexed sums

Let {S_n, n ≥ 1} be partial sums of independent identically distributed random variables. The almost sure version of CLT is generalized on the case of randomly indexed sums {S_Nn, n ≥ 1}, where {N_n, n ≥ 1} is a sequence of positive integer‐valued random variables independent of {S_n, n ≥ 1}. The affects of nonrandom centering and norming are considered too (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An almost sure central limit theorem for self-normalized weighted sums

Let X, X1 , X2 , ··· be a sequence of nondegenerate i.i.d. random variables with zero means, which is in the domain of attraction of the normal law. Let {a ni , 1≤i≤n, n≥1} be an array of real numbers with some suitable conditions. In this paper, we show that a central limit theorem for self-normalized weighted sums holds. We also deduce a version of ASCLT for self-normalized weighted sums.     

A FUNCTIONAL CENTRAL LIMIT THEOREMFOR NEGATIVELY ASSOCIATED SEQUENCE

Let(x1,j≥1)be a sequence of negatively associated random variables with ex1=o,ex^21＜∞.in this paper a functional central limit theorem for negatively associated random variables under some conditions withbout stationarity is proved which is the same as the results for positively associated random variables.     

Almost sure central limit theorem for partial sums and maxima

Let X, X₁, X₂, … be i.i.d. random variables with nondegenerate common distribution function F, satisfying EX = 0, EX² = 1. Let X_i and M_n = max{X_i, 1 ≤ i ≤ n }. Suppose there exists constants a_n > 0, b_n ∈ R and a nondegenrate distribution G (y) such that Then, we have almost surely, where f (x, y) denotes the bounded Lipschitz 1 function and Φ(x) is the standard normal distribution function (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Central limit theorem for quadratic forms for sparse tables

Sufficient conditions for asymptotic normality for quadratic forms in {n_t − np_t} are given, where {n_t} are the observed counts with expected cell means {np_t}. The main result is used to derive asymptotic distributions of many statistics including the Pearson's chi-square.     

The central limit theorem for stationary associated sequences

We study the problem of convergence in distribution of a suitably normalized sum of stationary associated random variables. We focus on the infinite variance case. New results are announced. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the central limit theorem for modulus trimmed sums

We prove a functional central limit theorem for modulus trimmed i.i.d. variables in the domain of attraction of a nonnormal stable law. In contrast to the corresponding result under ordinary trimming, our CLT contains a random centering factor which is inevitable in the nonsymmetric case. The proof is based on the weak convergence of a two-parameter process where one of the parameters is time and the second one is the fraction of truncation.     

A functional central limit theorem for a class of urn models

We construct an independent increments Gaussian process associated to a class of multicolor urn models. The construction uses random variables from the urn model which are different from the random variables for which central limit theorems are available in the two color case.     

Functional central limit theorem for random multilinear forms

Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 32, No. 2, pp. 175–186, April–June, 1992.     

The central limit theorem for lacunary series

In this paper, the central limit theorem for lacunary trigonometric series is proved. Two gap conditions by Erdos and Takahashi are extended and unified. The criterion for the Fourier character of lacunary series is also given.

     

A note on applications of the martingale central limit theorem to random permutations

A unified martingale approach is presented for establishing the asymptotic normality of some sequences of random variables. It is applied to the numbers of inversions, rises, and peaks, respectively, as well as the oscillation and the sum of consecutive pair products of a random permutation. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 323–332 (1997)     

An almost sure central limit theorem for products of sums of partial sums under association

Let be a strictly stationary positively or negatively associated sequence of positive random variables with EX₁=μ>0, and VarX₁=σ²<∞. Denote , and γ=σ/μ the coefficient of variation. Under suitable conditions, we show that

     

A central limit theorem for branching Brownian motion with random immigration

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law,where the immigration is determined by another branching Brownian motion.The limit is a Gaussian random measure and the normalization is t3/4for d=3 and t1/2for d≥4,where in the critical dimension d=4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.     

The central limit theorem for stochastic processes II

The main result is that the necessary and sufficient conditions for the central limit theorem for centered, second-order processes given by Giné and Zinn⁽⁶⁾ can be obtained without any basic measurability condition. Furthermore we extend some of their results.     

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 the...     

Some estimates in theL p metric in the central limit theorem for additive functions

The convergence of the value distributions of a normalized sequence of strongly additive arithmetic functions to the standard normal law in theL _p metric is considered. An asymptotic formula for the variance is obtained. In both cases, the remainders are expressed in terms of the third absolute moment of the additive function. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 74–80, January–March, 1999. Translated by A. Mačiulis