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.

中文翻译：

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投影准则下平稳随机场的淬火中心极限定理

受并非从平衡开始的随机演化的推动，Peligrad 和 Volný (J Theor Probab, 2018. arXiv:1802.09106) 在最近的一项工作中表明，中心极限定理 (CLT) 对平稳正交-鞅随机场成立从一个固定的过去轨迹开始。在本文中，我们研究了这种类型的行为，也称为淬火 CLT，用于一类大于正鞅的随机场。我们在投影标准方面强加了充分条件，在该标准下，平稳随机场的部分和允许正交鞅近似。更准确地说，充分条件是汉南射影类型的。我们还讨论了淬灭 CLT 功能形式的某些方面。作为应用程序，