Laws of the iterated logarithm for partial sum processes indexed by functions
For \(d\in N\), let \(\{X_ j\), \(j\in N^ d\}\) be an array of independent copies of a random variable X with E X\(=0\), E \(X^ 2=1\). For any Borel set \(A\subset [0,1)^ d=I\) define the random measure \(S_ n(A,X)=\sum_{j\in nA}X_ j\). Further if f is a measurable function on I one can define \[ S_ n(f,X)=\int_{I}f(t)S_ n(dt,X). \] We say that X satisfies the bounded law of the iterated logarithm for a class F of measurable functions if \[ \{(2n \log \log n)^{-1/2}S_ n(f,X)\} \] is a bounded subset of \(\ell^{\infty}(F)\). The notion of the compact LIL can be defined similarly. The author gives conditions on F implying that X satisfies the bounded resp. the compact LIL. Note that if \(F=\{1_{[0,t)}\), \(0\leq t\leq 1\}\) we obtain the classical LIL as special case.
scientific article; zbMATH DE number 4098393
scientific article