On the Quenched Central Limit Theorem for Stationary Random Fields Under Projective Criteria
Sur le Théorème Central Limite Quenched pour les champs stationnaires satisfaisant une condition projective du type Hannan
Résumé
Motivated by random evolutions which do not start from equilibrium, in a recent work, Peligrad and Volný (2018) 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 CLT's and their functional form for linear and nonlinear random fields with independent innovations.
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