Almost sure behavior of tail series in functional spaces
Let \(I_ n=\sum^{\infty}_{i=n}X_ iS_ i\), \(S_ i(t)=\int^{t}_{0}\phi_ i(u)du\), \(0\leq t\leq 1\), where \((X_ n)\) is a sequence of i.i.d. random variables and \((\phi_ n)\) is a complete orthonormal system in \(L_ 2(0,1)\). If \((X_ n)\) is Gaussian and E \(X_ 1=0\), E \(X^ 2_ 1=1\), it is well known that \(I_ 1\) converges a.s. in C(0,1), \(L_ p(0,1)\) (p\(\geq 1)\), and Lip \(\alpha\) \((0<\alpha <1/2)\) to Brownian motion on [0,1] [cf. \textit{V. V. Buldygin}, Convergence of random elements in topological spaces. (1980; Zbl 0512.60009)]. In this paper, results of strong law of large numbers type and iterated logarithm type are proved for \(\| I_ n\|\) where \(\|.\|\) denotes the norm in any of the spaces mentioned above. In detail there are studied the case that \((\phi_ n)\) is the Haar system and that \((X_ n)\) is non-Gaussian. This extends work by \textit{P. Hall} [Math. Proc. Camb. Philos. Soc. 92, 547-562 (1982; Zbl 0507.60078)] who proved similar results for the system \((\phi_ n)\) of trigonometric functions.
scientific article; zbMATH DE number 4159765