In this paper, we give sufficient conditions for the almost sure central limit theorem started at a point, known under the name of quenched central limit theorem. This is achieved by using a new idea ...of conditioning with respect to both the past and the future of the Markov chain. As applications, we provide a new sufficient projective condition for the quenched CLT.
In this paper we present a Bernstein-type tail inequality for the maximum of partial sums of a weakly dependent sequence of random variables that is not necessarily bounded. The class considered ...includes geometrically and subgeometrically strongly mixing sequences. The result is then used to derive asymptotic moderate deviation results. Applications are given for classes of Markov chains, iterated Lipschitz models and functions of linear processes with absolutely regular innovations.
In this paper we obtain the central limit theorem for triangular arrays of non-homogeneous Markov chains under a condition imposed to the maximal coefficient of correlation. The proofs are based on ...martingale techniques and a sharp lower bound estimate for the variance of partial sums. The results complement an important central limit theorem of Dobrushin based on the contraction coefficient.
Motivated by random evolutions which do not start from equilibrium, in a recent work, Peligrad and Volný (J Theor Probab, 2018.
arXiv:1802.09106
) showed that the central limit theorem (CLT) holds ...for stationary ortho-martingale random fields when they are started from a fixed past trajectory. In this paper, we study this type of behavior, also known under the name of quenched CLT, for a class of random fields larger than the ortho-martingales. We impose sufficient conditions in terms of projective criteria under which the partial sums of a stationary random field admit an ortho-martingale approximation. More precisely, the sufficient conditions are of the Hannan’s projective type. We also discuss some aspects of the functional form of the quenched CLT. As applications, we establish new quenched CLTs and their functional form for linear and nonlinear random fields with independent innovations.
In this paper, we study the central limit theorem and its functional form for random fields which are started not from their equilibrium, but rather under the measure conditioned by the past sigma ...field. The initial class considered is that of orthomartingales and then the result is extended to a more general class of random fields by approximating them, in some sense, with an orthomartingale. We construct an example which shows that there are orthomartingales which satisfy the CLT but not its quenched form. This example also clarifies the optimality of the moment conditions used for the validity of our results. Finally, by using the so-called orthomartingale-coboundary decomposition, we apply our results to linear and nonlinear random fields.
In this paper we study the central limit theorem for additive functionals of stationary Markov chains with general state space by using a new idea involving conditioning with respect to both the past ...and future of the chain. Practically, we show that any additive functionals of a stationary and totally ergodic Markov chain with var(Sn)∕n uniformly bounded, satisfies a n−central limit theorem with a random centering. We do not assume that the Markov chain is irreducible and aperiodic. However, the random centering is not needed if the Markov chain satisfies stronger forms of ergodicity. In absence of ergodicity the convergence in distribution still holds, but the limiting distribution might not be normal.
The goal of this paper is to indicate a new method for constructing normal confidence intervals for the mean, when the data is coming from stochastic structures with possibly long memory, especially ...when the dependence structure is not known or even the existence of the density function. More precisely we introduce a random smoothing suggested by the kernel estimators for the regression function. The normal confidence intervals are constructed under the sole condition that the sequence is ergodic and has finite second moments and a mild condition on the sample variance. Applications are presented for linear processes and reversible Markov chains with long memory.
•Our estimator satisfies the CLT and the functional CLT under a mild condition on the bandwidths sequence.•We propose confidence intervals for the mean under unquantified dependence.•We propose rules to choose the bandwidths sequence necessary in applications for confidence intervals.•No need to estimate the memory parameter in the case of long-memory processes.•This estimator provides striking results as shown in the simulation section.
We prove a central limit theorem for strictly stationary random fields under a sharp projective condition. The assumption was introduced in the setting of random sequences by Maxwell and Woodroofe. ...Our approach is based on new results for triangular arrays of martingale differences, which have interest in themselves. We provide as applications new results for linear random fields and nonlinear random fields of Volterra-type.
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