On convergence of moving average series of martingale differences fields taking values in Banach spaces

Ta Cong Son, Dang Hung Thang and Le Van Dung

Communications in Statistics - Theory and Methods, 2018, vol. 47, issue 22, 5590-5603

Abstract: Let {Xn,Fn;n∈Zd}$\lbrace X_{\bf n}, \mathcal {F}_{\bf n}; {\bf n}\in \mathbb {Z}^d\rbrace$ be a field of martingale differences taking values in a Banach space, {an;n∈Zd}$\lbrace a_{\bf n}; {\bf n} \in \mathbb {Z}^d\rbrace$ is an absolutely summable field of real numbers such that the moving average series Zk=∑i∈ZdaiXi+k$Z_{\bf k}=\sum _{{\bf i}\in \mathbb {Z}^d} a_{{\bf i}}X_{\bf i+k}$ converges almost surely. Let Tn = ∑k⪰nZk be the tail corresponding series which converges to 0 a.s. The paper provides conditions under which bnTn→0a.s.$b_{\bf n}T_{\bf n}\rightarrow 0\quad \text{a.s.}$ and P(supk⪰nbk∥Tk∥>ϵ)=o(1|n|1-α)asn→∞$P(\sup \limits _{ \bf k\succeq n} b_{\bf k}\Vert T_{\bf k}\Vert > \epsilon )=o(\dfrac{1}{|{\bf n}|^{1-\alpha } }) \ \mbox{as}\ {\bf n} \rightarrow \infty$ for every field of positive constants {bn, n⪰1} such that bn ⩽ bm for all n⪯m. These results are applied to obtain some results about the convergence of Quadratic Chaos.

Date: 2018
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DOI: 10.1080/03610926.2017.1397172

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