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On the Quenched Functional Central Limit Theorem for Stationary Random Fields under Projective Criteria

Lucas Reding, Na Zhang

Abstract

In this work we study and establish some quenched functional Central Limit Theorems (CLTs) for stationary random fields under a projective criteria. These results are functional generalizations of the theorems obtained by Zhang et al. (2020) and of the quenched functional CLTs for ortho-martingales established by Peligrad and Voln{ý} (2020) to random fields satisfying a Hannan type projective condition. In the work of Zhang et al. (2020), the authors have already proven a quenched functional CLT however the assumptions were not optimal as they require the existence of a 2 + $δ$-moment. In this article, we establish the results under weaker assumptions, namely we only require an Orlicz space condition to hold. The methods used to obtain these generalizations are somewhat similar to the ones used by Zhang et al. (2020) but we improve on them in order to obtain results within the functional framework. Moreover, a Rosenthal type inequality for said Orlicz space is also derived and used to obtain a sufficient condition analogous to that of Theorem 4.4 in the work of Zhang et al. (2020). Finally, we apply our new results to derive some quenched functional CLTs under weak assumptions for a variety of stochastic processes.

On the Quenched Functional Central Limit Theorem for Stationary Random Fields under Projective Criteria

Abstract

In this work we study and establish some quenched functional Central Limit Theorems (CLTs) for stationary random fields under a projective criteria. These results are functional generalizations of the theorems obtained by Zhang et al. (2020) and of the quenched functional CLTs for ortho-martingales established by Peligrad and Voln{ý} (2020) to random fields satisfying a Hannan type projective condition. In the work of Zhang et al. (2020), the authors have already proven a quenched functional CLT however the assumptions were not optimal as they require the existence of a 2 + -moment. In this article, we establish the results under weaker assumptions, namely we only require an Orlicz space condition to hold. The methods used to obtain these generalizations are somewhat similar to the ones used by Zhang et al. (2020) but we improve on them in order to obtain results within the functional framework. Moreover, a Rosenthal type inequality for said Orlicz space is also derived and used to obtain a sufficient condition analogous to that of Theorem 4.4 in the work of Zhang et al. (2020). Finally, we apply our new results to derive some quenched functional CLTs under weak assumptions for a variety of stochastic processes.

Paper Structure

This paper contains 11 sections, 17 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Framework and results
  3. Functional CLT over cubic regions
  4. Functional CLT over rectangular regions
  5. Proofs of the results
  6. Examples
  7. Linear field with independent innovations
  8. Nonlinearity: the case of Volterra fields
  9. Hölderian function of a linear field
  10. Weakly Dependent Processes
  11. Appendix

Key Result

Theorem \oldthetheorem

Assume that $(X_{\bm{n}})_{\bm{n} \in {\mathbb{Z}}^d}$ is defined by Xdef and that the filtration $(\mathcal{F}_{\bm{n}})_{\bm{n} \in {\mathbb{Z}}^d}$ given by Fdef is commuting. Also, assume that one of the transformations $T_i, 1 \le i \le d,$ is ergodic and that Then, for ${\mathbb{P}}$-almost all $\omega \in \Omega$, where $\sigma ^2 := {\mathbb{E}[D_{\bm{0}}^2]}$ with $D_{\bm{0}} = \sum_{\b

Theorems & Definitions (22)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem: Main Lemma
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 12 more