scholarly journals Asymptotic distribution with random indices for linear processes

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3925-3935
Author(s):  
Yu Miao ◽  
Qinghui Gao ◽  
Shuili Zhang
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Linear Process ◽  
Partial Sums ◽  
Real Numbers ◽  
Linear Processes ◽  
The Central Limit Theorem ◽  
Dependent Sequence ◽  
Random Indices

In this paper, we consider the following linear process Xn = ?? i=-? Ci?n-i, n ? Z, and establish the central limit theorem of the randomly indexed partial sums Svn := X1 +... + Xvn, where {ci,i?Z} is a sequence of real numbers, {?n,n?Z} is a stationary m-dependent sequence and {vn;n?1} is a sequence of positive integer valued random variables. In addition, in order to show the main result, we prove the central limit theorems for randomly indexed m-dependent random variables, which improve some known results.

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.

scholarly journals Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

2021 ◽  
Vol 36 (2) ◽  
pp. 243-255
Author(s):  
Wei Liu ◽  
Yong Zhang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Invariance Principle ◽  
Random Variables ◽  
Probability Measures ◽  
Linear Processes ◽  
The Central Limit Theorem

AbstractIn this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s central limit theorem and invariance principle to the case where probability measures are no longer additive.

scholarly journals An expansion related to the central limit theorem

1969 ◽  
Vol 10 (1-2) ◽  
pp. 219-230
Author(s):  
C. R. Heathcote
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Partial Sums ◽  
The Central Limit Theorem ◽  
Independent And Identically Distributed

Let X1, X2,…be independent and identically distributed non-lattice random variables with zero, varianceσ2<∞, and partial sums Sn = X1+X2+…+X.

scholarly journals On Feller's criterion for the law of the iterated logarithm

1994 ◽  
Vol 17 (2) ◽  
pp. 323-340 ◽  
Author(s):  
Deli Li ◽  
M. Bhaskara Rao ◽  
Xiangchen Wang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Uniform Estimate ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

Combining Feller's criterion with a non-uniform estimate result in the context of the Central Limit Theorem for partial sums of independent random variables, we obtain several results on the Law of the Iterated Logarithm. Two of these results refine corresponding results of Wittmann (1985) and Egorov (1971). In addition, these results are compared with the corresponding results of Teicher (1974), Tomkins (1983) and Tomkins (1990)

Linear processes are nearly Gaussian

1967 ◽  
Vol 4 (2) ◽  
pp. 313-329 ◽  
Author(s):  
C. L. Mallows
Keyword(s):  
Distribution Function ◽  
Continuous Distribution ◽  
Random Variables ◽  
Linear Process ◽  
Real Numbers ◽  
Linear Processes ◽  
Absolutely Continuous ◽  
Restricted Sense ◽  
Finite Set ◽  
Third Moment

Let U denote the set of all integers, and suppose that Y = {Yu; u ∈ U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.) G (·). Let a = {au; u ∈ U} be a sequence of real numbers with Σuau2 = 1. Then Xu = ΣwawYu–w defines a stationary linear process X = {Xu; u ɛ U} with E(Xu) = 0, E(Xu2) = 1 for u ∊ U. Let F(·) be the d.f. of X0. We prove that if maxu |au| is small, then (i) for each w, Xw is close to Gaussian in the sense that ∫∞−∞(F(y) − Φ(y))2dy ≦ g maxu |au | where Φ(·) is the standard Gaussian d.f., and g depends only on G(·); (ii) for each finite set (w1, … wn), (Xw1, … Xwn) is close to Gaussian in a similar sense; (iii) the process X is close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu|au| to the bandwidth of the filter a is studied.

Almost sure limit theorems for random allocations

2005 ◽  
Vol 42 (2) ◽  
pp. 173-194
Author(s):  
István Fazekas ◽  
Alexey Chuprunov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Domain ◽  
Almost Sure Limit Theorem ◽  
The Central Limit Theorem ◽  
Almost Sure Limit Theorems ◽  
Poisson Limit

Almost sure limit theorems are presented for random allocations. A general almost sure limit theorem is proved for arrays of random variables. It is applied to obtain almost sure versions of the central limit theorem for the number of empty boxes when the parameters are in the central domain. Almost sure versions of the Poisson limit theorem in the left domain are also proved.

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.

Linear processes are nearly Gaussian

1967 ◽  
Vol 4 (02) ◽  
pp. 313-329 ◽  
Author(s):  
C. L. Mallows
Keyword(s):  
Distribution Function ◽  
Continuous Distribution ◽  
Random Variables ◽  
Linear Process ◽  
Real Numbers ◽  
Linear Processes ◽  
Absolutely Continuous ◽  
Restricted Sense ◽  
Finite Set ◽  
Third Moment

LetUdenote the set of all integers, and suppose thatY= {Yu;u∈U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.)G(·). Leta= {au;u∈U} be a sequence of real numbers with Σuau2= 1. ThenXu= ΣwawYu–wdefines a stationary linear processX= {Xu; u ɛ U} withE(Xu) = 0,E(Xu2) = 1 foru∊U. LetF(·) be the d.f. ofX0.We prove that if maxu|au| is small, then (i) for eachw, Xwis close to Gaussian in the sense that ∫∞−∞(F(y) − Φ(y))2dy≦gmaxu|au| where Φ(·) is the standard Gaussian d.f., andgdepends only onG(·); (ii) for each finite set (w1, …wn), (Xw1, …Xwn) is close to Gaussian in a similar sense; (iii) theprocess Xis close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu|au| to the bandwidth of the filterais studied.

Central Limit Theorems for Dependent Heterogeneous Random Variables

1997 ◽  
Vol 13 (3) ◽  
pp. 353-367 ◽  
Author(s):  
Robert M. de Jong
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Martingale Difference ◽  
Dependent Random Variables ◽  
The Central Limit Theorem ◽  
Dependence Conditions

This paper presents central limit theorems for triangular arrays of mixingale and near-epoch-dependent random variables. The central limit theorem for near-epoch-dependent random variables improves results from the literature in various respects. The approach is to define a suitable Bernstein blocking scheme and apply a martingale difference central limit theorem, which in combination with weak dependence conditions renders the result. The most important application of this central limit theorem is the improvement of the conditions that have to be imposed for asymptotic normality of minimization estimators.

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