scholarly journals Almost sure convergence for self-normalized products of sums of partial sums of ρ¯-mixing sequences

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2471-2488
Author(s):  
Qunying Wu ◽  
Yuanying Jiang
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Almost Sure Convergence ◽  
Stationary Sequence ◽  
Central Limit Theorems ◽  
The Self ◽  
Partial Sums ◽  
Independent Case ◽  
Products Of Sums ◽  
Mixing Sequences

Let X,X1,X2,... be a stationary sequence of ??-mixing positive random variables. A universal result in the area of almost sure central limit theorems for the self-normalized products of sums of partial sums (?kj =1(Tj/(j(j+1)?/2)))?=(?Vk) is established, where: Tj = ?ji=1 Si,Si = ?i k=1 Xk,Vk = ??ki=1 X2i,? = EX, ? > 0. Our results generalize and improve those on almost sure central limit theorems obtained by previous authors from the independent case to ??-mixing sequences and from partial sums case to self-normalized products of sums of partial sums.

scholarly journals On the random functional central limit theorems with almost sure convergence for subsequences

2012 ◽  
Vol 45 (2) ◽  
Author(s):  
Zdzisław Rychlik ◽  
Konrad S. Szuster
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Central Limit Theorems ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Random Sum ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

AbstractIn this paper we present functional random-sum central limit theorems with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related ‘logarithmic’ limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random-sum central limit theorems for subsequences.

scholarly journals Almost Sure Limit Theorems for Multivariate General Standard Normal Sequences and Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Zhicheng Chen ◽  
Xinsheng Liu
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Almost Sure Convergence ◽  
Central Limit Theorems ◽  
Random Vectors ◽  
Standard Normal ◽  
Almost Sure Limit Theorems ◽  
General Standard

Under suitable conditions, the almost sure central limit theorems for the maximum of general standard normal sequences of random vectors are proved. The simulation of the almost sure convergence for the maximum is firstly performed, which helps to visually understand the theorems by applying to two new examples.

Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach

1973 ◽  
Vol 5 (01) ◽  
pp. 119-137 ◽  
Author(s):  
D. J. Scott
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Functional Central Limit Theorem ◽  
Stationary Increments ◽  
Mixing Sequences ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.

Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach

1973 ◽  
Vol 5 (1) ◽  
pp. 119-137 ◽  
Author(s):  
D. J. Scott
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Functional Central Limit Theorem ◽  
Stationary Increments ◽  
Mixing Sequences ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.

Central limit theorems for logarithmic averages

2001 ◽  
Vol 38 (1-4) ◽  
pp. 79-96
Author(s):  
I. Berkes ◽  
L. Horváth ◽  
X. Chen
Keyword(s):  
Wiener Process ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Partial Sums ◽  
Asymptotic Results

We prove central limit theorems and related asymptotic results for where W is a Wiener process and Sk are partial sums of i.i.d. random variables with mean 0 and variance 1. The integrability and smoothness conditions made on f are optimal in a number of important cases.

Gordin's theorem and the periodogram

1976 ◽  
Vol 13 (02) ◽  
pp. 365-370 ◽  
Author(s):  
Holger Rootzén
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Stationary Sequence ◽  
Central Limit Theorems ◽  
Stationary Processes

In 1968 M. I. Gordin proved a very strong central limit theorem for stationary, ergodic sequences by means of approximation with martingales. In the present paper Gordin's theorem is generalized to cover also the periodogram of a stationary sequence, and the restriction of ergodicity is removed. It is noted that known central limit theorems for stationary processes can often be generalized to the periodogram by means of this result.

Almost sure central limit theorems for weighted dependent sequences

2019 ◽  
Vol 56 (2) ◽  
pp. 145-153
Author(s):  
Khurelbaatar Gonchigdanzan
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Triangular Array ◽  
Central Limit Theorems ◽  
Partial Sums ◽  
Real Numbers ◽  
Dependent Random Variables ◽  
Associated Sequences

Abstract Let {Xn: n ≧ 1} be a sequence of dependent random variables and let {wnk: 1 ≦ k ≦ n, n ≧ 1} be a triangular array of real numbers. We prove the almost sure version of the CLT proved by Peligrad and Utev [7] for weighted partial sums of mixing and associated sequences of random variables, i.e.

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