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Magda Peligrad

Publications and source records attributed to Magda Peligrad.

At least 19 recordsLinked to original sources
arXiv2022

Convergence of series of conditional expectations

This paper deals with rates of convergence in the strong law of large numbers, in the Baum-Katz form, for partial sums of Banach space valued random variables. The results are then applied to solve similar problems for weighted partial sums of conditional expectations. They are further used to treat partial sums of powers of a reversible Markov chain operator. The method of proof is based on martingale approximation. The conditions are expressed in terms moments of the individual summands.

math.PR↗
arXiv2022

On the quenched CLT for stationary Markov chains

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 conditions for the quenched CLT.

math.PR↗
arXiv2022

On the invariance principle for reversible Markov chains

In this paper, we investigate the functional central limit theorem 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 central limit theorem 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.

math.PR↗
arXiv2021

On the local limit theorems for lower psi-mixing Markov chains

In this paper we investigate the local limit theorem for additive functionals of nonstationary Markov chains that converge in distribution. We consider both the lattice and the non-lattice cases. The results are also new in the stationary setting and lead to local limit theorems linked to convergence to stable distributions. The conditions are imposed to individual summands and are expressed in terms of lower psi-mixing coefficients.

math.PR↗
arXiv2021

Limit theorems for linear random fields with innovations in the domain of attraction of a stable law

In this paper we study the convergence in distribution and the local limit theorem for the partial sums of linear random fields with i.i.d. innovations that have infinite second moment and belong to the domain of attraction of a stable law with index $0<\alpha\leq2$ under the condition that the innovations are centered if $1<\alpha\leq2$ and are symmetric if $\alpha=1$. We establish these two types of limit theorems as long as the linear random fields are well-defined, the coefficients are either absolutely summable or not absolutely summable.

math.PR↗
arXiv2020

A Local Limit Theorem for Linear Random Fields

In this paper, we establish a local limit theorem for linear fields of random variables constructed from independent and identically distributed innovations each with finite second moment. When the coefficients are absolutely summable we do not restrict the region of summation. However, when the coefficients are only square-summable we add the variables on unions of rectangle and we impose regularity conditions on the coefficients depending on the number of rectangles considered. Our results are new also for the dimension 1, i.e. for linear sequences of random variables. The examples include the fractionally integrated processes for which the results of a simulation study is also included.

math.PR↗
arXiv2020

On the local limit theorems for psi-mixing Markov chains

In this paper we investigate the local limit theorem for additive functionals of a nonstationary Markov chain with finite or infinite second moment. The moment conditions are imposed on the individual summands and the weak dependence structure is expressed in terms of some uniformly mixing coefficients.

math.PR↗
arXiv2020

On the CLT of additive functionals of Markov chains

In this paper we study the additive functionals of Markov chains via conditioning with respect to both past and future of the chain. We shall point out new sufficient projective conditions, which assure that the variance of partial sums of n consecutive random variables of a stationary Markov chain is linear in n. The paper also addresses the central limit theorem problem and is listing several open questions.

math.PR↗
arXiv2020

A new CLT for additive functionals of Markov chains

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 the variance of S_n divided by n uniformly bounded, satisfies a square root 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 an ergodicity the convergence in distribution still holds, but the limiting distribution might not be normal.

math.PR↗
arXiv2019

On the quenched CLT for stationary random fields under projective criteria

Motivated by random evolutions which do not start from equilibrium, in a recent work, Peligrad and Volný (2018) showed that the quenched CLT (central limit theorem) holds for ortho-martingale random fields. In this paper, we study the quenched CLT for a class of random fields larger than the ortho-martingales. To get the results, 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. As applications, we establish quenched CLT's for linear and nonlinear random fields with independent innovations.

math.PR↗
arXiv2019

Quenched invariance principles for orthomartingale-like sequences

In this paper we study the central limit theorem and its functional form for random fields which are not started 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.

math.PR↗
arXiv2018

Functional CLT for martingale-like nonstationary dependent structures

In this paper we develop non-stationary martingale techniques for dependent data. We shall stress the non-stationary version of the projective Maxwell-Woodroofe condition, which will be essential for obtaining maximal inequalities and functional central limit theorem for the following examples: nonstationary \r{ho}-mixing sequences, functions of linear processes with non-stationary innovations, quenched version of the functional central limit theorem for a stationary sequence, evolutions in random media such as a process sampled by a shifted Markov chain.

math.PR↗
arXiv2017

New robust confidence intervals for the mean under dependence

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. Applications are presented to linear processes and reversible Markov chains with long memory.

stat.ME↗
arXiv2017

Martingale approximations for random fields

In this paper we provide necessary and sufficient conditions for the mean square approximation of a random field with an ortho-martingale. The conditions are formulated in terms of projective criteria. Applications are given to linear and nonlinear random fields with independent innovations.

math.PR↗
arXiv2017

Central limit theorem for Fourier transform and periodogram of random fields

In this paper we show that the limiting distribution of the real and the imaginary part of the double Fourier transform of a stationary random field is almost surely an independent vector with Gaussian marginal distributions, whose variance is, up to a constant, the field's spectral density. The dependence structure of the random field is general and we do not impose any restrictions on the speed of convergence to zero of the covariances, or smoothness of the spectral density. The only condition required is that the variables are adapted to a commuting filtration and are regular in some sense. The results go beyond the Bernoulli fields and apply to both short range and long range dependence. They can be easily applied to derive the asymptotic behavior of the periodogram associated to the random field. The method of proof is based on new probabilistic methods involving martingale approximations and also on borrowed and new tools from harmonic analysis.

math.PR↗
arXiv2017

On the normal approximation for random fields via martingale methods

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 variables 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.

math.PR↗
arXiv2016

The limiting spectral distribution in terms of spectral density

For a large class of symmetric random matrices with correlated entries, selected from stationary random fields of centered and square integrable variables, we show that the limiting distribution of eigenvalue counting measure always exists and we describe it via an equation satisfied by its Stieltjes transform. No rate of convergence to zero of correlations is imposed, therefore the process is allowed to have long memory. In particular, if the symmetrized matrices are constructed from stationary Gaussian random fields which have spectral density, the result of this paper gives a complete solution to the limiting eigenvalue distribution. More generally, for matrices whose entries are functions of independent identically distributed random variables the result also holds.

math.PR↗