Some laws of the iterated logarithm for two parameter martingales
The author obtains some laws of the iterated logarithm for two parameter martingales \(X=\{X_t: t\in N^2\}\) where \(N\) denotes the set of integers. Put \({\mathcal F}_t= \sigma\{X_s:s\leq t\}\), \(t\in N^2\), where the inequality \(s\leq t\) is meant to hold componentwise, and let \({\mathcal F}_1^{(m)}(0-)\) \((m=0,1,\dots)\) denote the \(\sigma\)-algebra generated by the union of all \(\sigma\)-algebras \({\mathcal F}_s\) where \(s=(s_1,s_2)\in N^2\) is such that \(s_1<0\) or \(0\leq s_1\leq m\), \(s_2<0\). In order to formulate one of the main results let \(X\) be strictly stationary with \(E[X_0^2]=1\), \(E[| X_0|^q] <\infty\) for some \(q>2\) and \(E[X_0\mid {\mathcal F}_1^{(m)} (0-)]=0\) a.s., \(m=0,1,\dots\) Finally put \[ \zeta_n=\biggl(4| n|\log\bigl(\log| n|\bigr)\biggr)^{-1/2} \sum_{1\leq t\leq n}X_t, \quad n\geq 1\;(n\in N^2), \] where \(| n|\) is the product of the absolute values of the nonzero components of \(n\). Then the following is proved: (i) If \(X\) is ergodic, the set of limit points of \((\zeta_n)\) (as \(n_1,n_2 \to\infty)\) is a.s. equal to \([-1,1]\). (ii) If \(X\) is strongly ergodic [in the sense of the author, Chin. Ann. Math., Ser. B 12, No. 4, 432-444 (1991; Zbl 0764.62075)], the set of limit points of \((\zeta_n)\) (as \(| n|\to\infty)\) is a.s. equal to \([-1,1]\).
scientific article; zbMATH DE number 1275646