Gaussian Approximation under Asymptotic Negative Dependence

2019 ◽  
pp. 277-302
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Gaussian Approximation ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Practical Applications ◽  
Functional Clt ◽  
Dependent Sequence ◽  
Associated Sequences ◽  
Maximal Moment

Here we introduce the notion of asymptotic weakly associated dependence conditions, the practical applications of which will be discussed in the next chapter. The theoretical importance of this class of random variables is that it leads to the functional CLT without the need to estimate rates of convergence of mixing coefficients. More precisely, because of the maximal moment inequalities established in the previous chapter, we are able to prove tightness for a stochastic process constructed from a negatively dependent sequence. Furthermore, we establish the convergence of the partial sums process, either to a Gaussian process with independent increments or to a diffusion process with deterministic time-varying volatility. We also provide the multivariate form of these functional limit theorems. The results are presented in the non-stationary setting, by imposing Lindeberg’s condition. Finally, we give the stationary form of our results for both asymptotic positively and negatively associated sequences of random variables.

scholarly journals Almost sure local central limit theorem for the product of some partial sums of negatively associated sequences

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Feng Xu ◽  
Binhui Wang ◽  
Yawen Hou
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
General Result ◽  
Random Variables ◽  
Partial Sums ◽  
Negatively Associated ◽  
Associated Sequences ◽  
Local Central Limit Theorem

AbstractThe almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem. Let $\{X_{k},k\geq 1\}${Xk,k≥1} be a strictly stationary negatively associated sequence of positive random variables. Under the regular conditions, we discuss an almost sure local central limit theorem for the product of some partial sums $(\prod_{i=1}^{k} S_{k,i}/((k-1)^{k}\mu^{k}))^{\mu/(\sigma\sqrt{k})}$(∏i=1kSk,i/((k−1)kμk))μ/(σk), where $\mathbb{E}X_{1}=\mu$EX1=μ, $\sigma^{2}={\mathbb{E}(X_{1}-\mu)^{2}}+2\sum_{k=2}^{\infty}\mathbb{E}(X_{1}-\mu)(X_{k}-\mu)$σ2=E(X1−μ)2+2∑k=2∞E(X1−μ)(Xk−μ), $S_{k,i}=\sum_{j=1}^{k}X_{j}-X_{i}$Sk,i=∑j=1kXj−Xi.

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.

Application to the Uniform Laws of Large Numbers for Dependent Processes

2019 ◽  
pp. 438-447
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Random Variables ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Single Function ◽  
Strongly Mixing ◽  
Mixing Coefficients ◽  
Mixing Sequences ◽  
Dependent Processes

As mentioned in Chapter 5, one of the most powerful techniques to derive limit theorems for partial sums associated with a sequence of random variables which is mixing in some sense is the coupling of the initial sequence by an independent one having the same marginal. In this chapter, we shall see how the coupling results mentioned in Section 5.1.3 are very useful to derive uniform laws of large numbers for mixing sequences. The uniform laws of large numbers extend the classical laws of large numbers from a single function to a collection of such functions. We shall address this question for sequences of random variables that are either absolutely regular, or ϕ‎-mixing, or strongly mixing. In all the obtained results, no condition is imposed on the rates of convergence to zero of the mixing coefficients.

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 Convergence Rates for Probabilities of Moderate Deviations for Multidimensionally Indexed Random Variables

2009 ◽  
Vol 2009 ◽  
pp. 1-15
Author(s):  
Dianliang Deng
Keyword(s):  
Convergence Rates ◽  
Broad Class ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Moderate Deviations ◽  
Partial Sums ◽  
Tail Probabilities ◽  
Class Of Functions ◽  
Equivalent Conditions

Let{X,Xn¯;n¯∈Z+d}be a sequence of i.i.d. real-valued random variables, andSn¯=∑k¯≤n¯Xk¯,n¯∈Z+d. Convergence rates of moderate deviations are derived; that is, the rates of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of the series∑n¯b(n¯)ψ2(a(n¯))P{|Sn¯|≥a(n¯)ϕ(a(n¯))}, wherea(n¯)=n11/α1⋯nd1/αd,b(n¯)=n1β1⋯ndβd,ϕandψare taken from a broad class of functions. These results generalize and improve some results of Li et al. (1992) and some previous work of Gut (1980).

Reversible Markov Chains

2019 ◽  
pp. 405-427
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Gaussian Approximation ◽  
Building Blocks ◽  
Partial Sums ◽  
Reversible Markov Chain ◽  
Maximal Inequalities ◽  
Functional Clt ◽  
Reversible Markov Chains ◽  
Functional Central Limit

This chapter is dedicated to the Gaussian approximation of a reversible Markov chain. Regarding this problem, the coefficients of dependence for reversible Markov chains are actually the covariances between the variables. We present here the traditional form of the martingale approximation including forward and backward martingale approximations. Special attention is given to maximal inequalities which are building blocks for the functional limit theorems. When the covariances are summable we present the functional central limit theorem under the standard normalization √n. When the variance of the partial sums are regularly varying with n, we present the functional CLT using as normalization the standard deviation of partial sums. Applications are given to the Metropolis–Hastings algorithm.

scholarly journals On one-sided boundedness of normed partial sums

1980 ◽  
Vol 21 (3) ◽  
pp. 373-391 ◽  
Author(s):  
R. A. Maller
Keyword(s):  
Relative Stability ◽  
Regular Variation ◽  
Random Variables ◽  
Partial Sums ◽  
Sufficient Condition ◽  
Iterated Logarithm ◽  
Dependent Sequence ◽  
General Sufficient Condition ◽  
The One ◽  
Independent And Identically Distributed

This paper gives a very general sufficient condition for the existence of constants B(n), C(n) for which either almost surely or almost surely, where Sn = X1 + X2 + … + Xn and Xi are independent and identically distributed random variables. The theorem is closely connected with results of Klass and Teicher on the one-sided boundedness of Sn, with the relative stability of Sn, and with a generalised law of the iterated logarithm due to Kesten. For non negative Xi the sufficient condition is shown to be necessary, and the results are partially generalised to the case when Xi form a stationary m-dependent sequence. Some connections with a generalised type of regular variation and with domains of partial attraction are also noted.

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