Variance Of Partial Sums Research Articles

Estimation of the limit variance for sums under a new weak dependence condition

ABSTRACTWe prove a self-normalized central limit theorem for a mixing class of processes introduced in Kacem M, Loisel S, Maume-Deschamps V. [Some mixing properties of conditionally independent processes. Commun Statist Theory Methods. 2016;45:1241–1259]. This class is larger than more classical strongly mixing processes and thus our result is more general than [Peligrad M, Shao QM. Estimation of the variance of partial sums for ρ-mixing random variables. J Multivar Anal. 1995;52:140–157; Shi S. Estimation of the variance for strongly mixing sequences. Appl Math J Chinese Univ. 2000;15(1):45–54] ones. The fact that some conditionally independent processes satisfy this kind of mixing properties motivated our study. We investigate the weak consistency as well as the asymptotic normality of the estimator of the variance that we propose.

QUENCHED CENTRAL LIMIT THEOREMS FOR A STATIONARY LINEAR PROCESS

We find a sufficient condition under which a central limit theorem for a stationary linear process is quenched. We find a stationary linear process szatisfying the Maxwell-Woodroofe condition for which the variances of partial sums are o(n), there is a CLT with a convergence towards N(0,1) when dividing by standard deviation of the partial sums, and the CLT is not quenched. The weak invariance principle does not hold.

On the invariance principle for reversible Markov chains

Abstract In this paper we investigate the functional central limit theorem (CLT) for stochastic processes associated to partial sums of additive functionals of reversible Markov chains with general spate space, under the normalization standard deviation of partial sums. For this case, we show that the functional CLT is equivalent to the fact that the variance of partial sums is regularly varying with exponent 1 and the partial sums satisfy the CLT. It is also equivalent to the conditional CLT.

Asymptotic variance of stationary reversible and normal Markov processes

We obtain necessary and sufficient conditions for the regularvariation of the variance of partial sums of functionals of discrete and continuous-time stationary Markov processes with normal transition operators. We also construct a class of Metropolis-Hastings algorithms which satisfy a central limit theorem and invariance principle when the variance is not linear in n.

The variance of partial sums of strong near-epoch dependent variables

In this paper, we discuss the lower bound for the variance of partial sums of strong near-epoch dependent variables under quite weak conditions. By suitably applying this result, we derive the limit behavior of the variance of the partial sums, which is especially useful in studying the large sample properties for econometric time series models with strong near-epoch dependent innovations.

Strong mixing coefficients for non-commutative Gaussian processes

Bounds for non-commutative versions of two classical strong mixing coefficients for q q -Gaussian processes are found in terms of the angle between the underlying Hilbert spaces. As a consequence, we construct a ψ \psi -mixing q q -Gaussian stationary sequence with growth conditions on variances of partial sums. If classical processes with analogous properties were to exist, they would provide a counter-example to the Ibragimov conjecture.

Self-normalized central limit theorem and estimation of variance of partial sums for negative dependent random variables

Let {Xn, n≥1} be a stationary LNQD or NA sequence satisfying EX1 = μ, EX12<∞ and (Var Sn)/n→σ2 as n→∞. In this paper a class of self-normalized central limit theorems and estimators of Var Sn are studied. The weak and strong consistency of the estimators of Var Sn are presented.