On the Quenched Central Limit Theorem for Stationary Random Fields Under Projective Criteria
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On the Quenched Central Limit Theorem for Stationary Random Fields Under Projective Criteria Na Zhang1 · Lucas Reding2 · Magda Peligrad3 Received: 4 July 2019 / Revised: 4 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract 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. Keywords Random fields · Quenched central limit theorem · Ortho-martingale approximation · Projective criteria Mathematics Subject Classification (2010) 60G60 · 60F05 · 60G42 · 60G48 · 41A30
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Na Zhang [email protected] Lucas Reding [email protected] Magda Peligrad [email protected]
1
Department of Mathematics, Towson University, Towson, MD 21252-0001, USA
2
Université de Rouen Normandie, 76801 Saint-Étienne-du-Rouvray, France
3
University of Cincinnati, PO box 210025, Cincinnati, OH 45221-0025, USA
123
Journal of Theoretical Probability
1 Introduction An interesting problem, with many practical applications, is to study limit theorems for processes conditioned to start from a fixed past trajectory. This problem is difficult, since the stationary processes started from a fixed past trajectory, or from a point, are no longer stationary. Furthermore, the validity of a limit theorem is not enough to assure that the convergence still holds when the process is not started from its equilibrium. This type of convergence is also known under the name of almost sure conditional limit theorem or the quenched limit theorem. The issue of the quenched CLT for stationary processes has been widely explored for the last few decades. Among many others, we mention papers by Derriennic and Lin [11], Cuny and Peligrad [5], Cuny and Volný [8], Cuny and Merlevède [6], Volný and Woodroofe [29], Barrera et al. [1]. Some of these results were surveyed in Peligrad [21]. A random field consists of multi-indexed random variables (X u )u∈Z d , where d is a positive integer. The main difficulty, when analyzing the asymptotic properties of random fields, is the fact that the future and the past do not have a unique interpretation. To compensate for the lack of ordering of the filtration, it is customary to use the notion of commuting filtrations. Traditionally, this ki
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