Deviation inequality for Banach-valued orthomartingales (Q6596204)

Consider \(\mathbb{Z}^d\) with the coordinatewise partial order: \(\mathbf{i}=(i_1,\dots,i_d) \preceq \mathbf{j} = (j_1,\dots,j_d)\) if \(i_l\leq j_l\) for every \(l\in\{1,\dots,d\}\). A collection of \(\sigma\)-algebras \((\mathcal{F}_{\mathbf{i}})_{\mathbf{i}\in\mathbb{Z}^d}\) is a completely commuting filtration if \(\mathcal{F}_{\mathbf{i}}\subset \mathcal{F}_{\mathbf{j}}\) for \(\mathbf{i}\preceq\mathbf{j}\) and \(\mathbb{E}\big(\mathbb{E}(\cdot| \mathcal{F}_{\mathbf{i}}) \big| \mathcal{F}_{\mathbf{j}}\big) = \mathbb{E}(\cdot| \mathcal{F}_{\min\{\mathbf{i}, \mathbf{j}\}})\) for all \(\mathbf{i}\), \(\mathbf{j}\) (here \(\min\{\mathbf{i}, \mathbf{j}\}\) is the coordinatewise minimum). A collection of random variables \((X_{\mathbf{i}})_{\mathbf{i}\in\mathbb{Z}^d}\) (with values in some separable Banach space) is an orthomartingale difference random field if the random variables \(X_{\mathbf{i}}\) are integrable, \(\mathcal{F}_{\mathbf{i}}\)-measurable, and \(\mathbb{E}( X_{\mathbf{i}}| \mathcal{F}_{\mathbf{i} - \mathbf{e}_l }) = 0\), where \(\mathbf{e}_l\), \(l\in\{1,\dots, d\}\), are standard unit vectors. For \(\mathbf{1} = (1,\dots, 1) \preceq \mathbf{n}=(n_1,\dots,n_d)\) set \(S_\mathbf{n} = \sum_{\mathbf{1} \preceq \mathbf{i} \preceq \mathbf{n}} X_{\mathbf{i}}\).\N\NUnder some assumptions on the geometry of the Banach space, the author provides a deviation inequality for \(\max_{\mathbf{1} \preceq \mathbf{n} \preceq \mathbf{N}} \| S_\mathbf{n} \|\). The proof is based on induction on the dimension \(d\). Some applications to regression models and laws of large numbers are discussed.\N\NThe abstract and introduction of this article contain a number of copy-editing errors (e.g. ``deviation inequality inequalities''), which would not be unexpected in a preprint, but -- in the reviewer's opinion -- is surprising in a published article and actually raises concerns about the thoroughness of the peer review and proofreading processes.