The law of the iterated logarithm for independent random variables with multidimensional indices
Let \(X_{\mathbf n}\), \({\mathbf n}=(n_ 1,\ldots,n_ d)\in{\mathcal N}^ d\), \(d\geq 1\), be a field of independent variables, \(a_{\mathbf n}\), \({\mathbf n}\in{\mathcal N}^ d\), be a field of real constants satisfying the so- called star-property and \(S_{\mathbf n}=\sum_{{\mathbf i}\leq{\mathbf n}}X_{{\mathbf i}}\) where \(\leq\) denotes the coordinate-wise ordering. The star-property covers a very broad class of normalizing constants \(a_{{\mathbf n}}\) including \(a_{{\mathbf n}}=|{\mathbf n}|^ \gamma\varphi(|{\mathbf n}|)\), \({\mathbf n}\in{\mathcal N}^ d\), where \(\varphi\) is slowly varying, \(\gamma>0\), and \(|{\mathbf n}|=n_ 1\cdots n_ d\), or \(a_{{\mathbf n}}\uparrow\infty\) for \(d=1\). The star property was introduced by the reviewer and \textit{R. Norvaisa} [Probab. Theory Relat. Fields 74, 241-253 (1987; Zbl 0586.60007)] in connection with the SLLN for rectangular sums. The authors prove results of BLIL-type assuming conditions on \(\sum_{{\mathbf i}\leq{\mathbf n}}X^ 2_{{\mathbf i}}/a_{{\mathbf n}}\). Theorem 1: If \((S_{{\mathbf n}}/\sqrt{2a_{{\mathbf n}}\log\log a_{{\mathbf n}}})\) is stochastically bounded and \[ \limsup_{{\mathbf n}\in{\mathcal N}^ d}\sum_{{\mathbf i}\leq{\mathbf n}}X^ 2_{{\mathbf i}}/a_{{\mathbf n}}<\infty \text{ a.s.,} \] then \[ \limsup_{{\mathbf n}\in{\mathcal N}^ d}| S_{{\mathbf n}}|/\sqrt{2a_{{\mathbf n}}\log\log a_{{\mathbf n}}}<\infty \text{ a.s.} \] Further results (Theorem 2 and Corollaries) give necessary and sufficient conditions for the SLLN and the BLIL with normalization \(a_{\mathbf n}\) reducing these problems to the condition \[ \sum_{{\mathbf n}\in{\mathcal N}^ d}P(| X_{{\mathbf n}}|\geq \varepsilon a_{{\mathbf n}})<\infty \] for every or some \(\varepsilon >0\), respectively. Theorem 3 is an analogue of Wittmann's LIL [see \textit{R. Wittmann}, Z. Wahrscheinlichkeitstheorie Verw. Geb. 68, 521-543 (1985; Zbl 0554.60040) for the case \(d=1\)]. Analogous results are proven for random variables taking values in Banach space. In the proofs the authors use the Gaussian randomization technique introduced by \textit{M. Ledoux} and \textit{M. Talagrand} [Ann. Probab. 16, No. 3, 1242-1264 (1988; Zbl 0662.60008.)].
scientific article; zbMATH DE number 89782