Partial Sums Of Linear Processes Research Articles

Limit distribution for products of sums of partial sums of linear processes generated by NSD sequence

Let be a negatively superadditive dependent (NSD) sequence of positive random variables. In this paper, under suitable conditions, we obtain the limit distribution for products of sums of partial sums of linear processes generated by

Limit theorems for linear processes generated by ρ-mixing sequence

Let be a strictly stationary ρ-mixing sequence of non degenerate random variables with zero means. In this paper, we study the central limit theorem and almost sure central limit theorem for self-normalized partial sums of linear processes generated by

Self-normalized limit theorems for linear processes generated by ρ-mixing innovations*

In this paper, we study the asymptotic behavior of the self-normalizer Vn2 for partial sums of linear processes generated by strictly stationary ρ-mixing innovations with infinite variance. Further, by using this we derive self-normalized versions of the CLT, the functional CLT, and the almost sure CLT for partial sums of the processes.

Nonlinear regressions with nonstationary time series

This paper develops asymptotic theory for a nonlinear parametric cointegrating regression model. We establish a general framework for weak consistency that is easy to apply for various nonstationary time series, including partial sums of linear processes and Harris recurrent Markov chains. We provide limit distributions for nonlinear least square estimators, extending the previous works. We also introduce endogeneity to the model by allowing the error to be serially dependent on and cross correlated with the regressors.

Functional Convergence of Linear Processes with Heavy-Tailed Innovations

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients, necessary and sufficient conditions for the finite dimensional convergence to an \(\alpha \)-stable Levy Motion are given. The conditions lead to new, tractable sufficient conditions in the case \(\alpha \le 1\). In the functional setting, we complement the existing results on \(M_1\)-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (Ann Probab 20:483–503, 1992) and improved by Louhichi and Rio (Electr J Probab 16(89), 2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the \(S\) topology, introduced by Jakubowski (Electr J Probab 2(4), 1997).

A CENTRAL LIMIT THEOREM FOR LINEAR PROCESSES UNDER LINEAR NEGATIVELY QUADRANT DEPENDENCE

Abstract. In this paper we establish a central limit theorem forweighted sums of Y n =P ni =1 a n;i X i , where fa n;i ;n 2 N; 1 • i • ng is an array of nonnegative numbers such that sup n‚ 1 P ni =1 a 2 n;i <1 , max 1 •i•n a n;i ! 0 and fX i ;i 2 Ng is a sequence of linearnegatively quadrant dependent random variables with EX i = 0and EX 2 i < 1 . Using this result we will obtain a central limittheorem for partial sums of linear processes. 1. IntroductionFor a sequence fa n ;n ‚ 1 g of real numbers the limit superior is de-flnedbyinf r‚ 1 sup n‚ a n andisdenotedbylimsup n!1 a n . Let fX n ;n ‚ 1 g be a sequence of random variables and fa n;k ;n ‚ 1 ; 1 • k • ng bean array of real numbers. The weighted sumsP nk =1 a n;k X k can play animportant role in various applied and theoretical problems, such as thoseof the least squares estimators(see Ka°es and Bhaskara Rao(1982)) andM-estimates(see Rao and Zhao(1992)) in linear models, the nonpara-metric regression estimators(see Priestley and Chao(1972)), etc. So thestudy of the central limit theorem is every important and signiflcant.Two random variables

Asymptotic Distribution for Products of Sums of Linear Processes Under Dependence

The products of partial sums of linear processes generated by sequences of positive random variables, which are either linear negative quadrant dependent or linear positive quadrant dependent, is asymptotically lognormal. This extends the earlier work on products of partial sums of random variables.

Asymptotics of partial sums of linear processes with changing memory parameter*

We study the limit distribution of partial sums of nonstationary truncated linear process {Xt, t = 1,…, n} with long memory and changing memory parameter dt,n ∈ (0,∞). Two classes of linear processes are investigated, namely, (i) the class of FARIMA-type truncated moving averages with time-varying fractional integration parameter and (ii) the class of time-varying fractionally integrated processes introduced in [A. Philippe, D. Surgailis, and M.-C. Viano, Invariance principle for a class of nonstationary processes with long memory, C. R. Acad. Sci., Paris, Ser. I, 342:269–274, 2006; A. Philippe, D. Surgailis, and M.-C. Viano, Time-varying fractionally integrated processes with nonstationary long memory, Theory Probab. Appl., 52:651–673, 2008]. The cases of fast-changing memory parameter (dt,n = dt does not depend on n) and slowly changing memory parameter (dt,n = d(t/n) for some function d(τ), τ ∈ [0, 1]) are discussed. In the case of fast-changing memory, the limit partial sums process is a type II fractional Brownian motion (fBm) with the Hurst parameter equal to the global maximum of (dt) for class (i) processes and to the mean value of (dt) for class (ii) processes. In the case of slowly changing memory, the limit of partial sums for both classes (i) and (ii) is degenerate and “localized” at the global maximum of the memory function d(∙); however, a nondegenerate limit of the partial sums process is shown to exist when time is suitably rescaled in the vicinity of the maximum point.

Functional central limit theorems for self-normalized partial sums of linear processes

We prove an invariance principle under self-normalization by blocks for linear processes with summable filters and i.i.d. innovations in the domain of attraction of the normal distribution.

Invariance principles for linear processes with application to isotonic regression

In this paper, we prove maximal inequalities and study the functional central limit theorem for the partial sums of linear processes generated by dependent innovations. Due to the general weights, these processes can exhibit long-range dependence and the limiting distribution is a fractional Brownian motion. The proofs are based on new approximations by a linear process with martingale difference innovations. The results are then applied to study an estimator of the isotonic regression when the error process is a (possibly long-range dependent) time series.

Asymptotic Independence of Distant Partial Sums of Linear Processes

We investigate the joint weak convergence (f.d.d. and functional) of the vector-valued process (U n (1) (τ), U n (2) (τ)) for τ ∈ [0, 1], where $$U_n^{(1)} (\tau )\;: = \;A_n^{ - 1} \;\Sigma _{t = 1}^{[n\tau ]} \;X_t $$ and $$U_n^{(2)} (\tau )\;: = \;A_n^{ - 1} \;\Sigma _{t = 1}^{[n\tau ]} \;X_{t + m} $$ are normalized partial-sum processes separated by a large lag m, m/n → ∞, and (X t , t ∈ ℤ) is a stationary moving-average process with i.i.d. (or martingale-difference) innovations having finite variance. We consider the cases where (X t ) is a process with long memory, short memory, or negative memory. We show that, in all these cases, as n → ∞ and m/n → ∞, the bivariate partial-sum process (U n (1) (τ), U n (2) (τ)) tends to a bivariate fractional Brownian motion with independent components. The result is applied to prove the consistency of certain increment-type statistics in moving-average observations.

A Random Functional Central Limit Theorem for Processes of Product Sums of Linear Processes Generated by Martingale Differences

A random functional central limit theorem is obtained for processes of partial sums and product sums of linear processes generated by non-stationary martingale differences. It develops and improves some corresponding results on processes of partial sums of linear processes generated by strictly stationary martingale differences, which can be found in [5].