On the one-sided logarithmic law for arrays (Q2721863)

Let \(\{X_{nk},\;k=1,\ldots,n,\;n=1,2\ldots\}\) be an array of i.i.d. random variables. Set \(S_n = \displaystyle\sum_{k=1}^n X_{nk}\). Assume \(EX_{11}=0\). Then (i) \(\displaystyle\limsup_{n \to \infty} S_n/(2n \log n)^{1/2} = 1\) a.s. iff (ii) \(E(X_{11}^{+})^4 (\log^{+} X_{11})^{-2} < \infty\) and \(EX_{11}^2 =1\), here \(x^{+} = \max (0,x)\), \(\log^{+} x = \log \max (e,x)\). Previously \textit{Y. C. Qi} [Bull. Aust. Math. Soc. 50, No. 2, 219-223 (1994; Zbl 0816.60030)] has given necessary and sufficient conditions for simultaneus validity of (i) and \(\displaystyle \liminf_{n \to \infty} S_n / (2n \log n)^{1/2} = -1\) a.s. The author conjectures that (i) holds iff \(EX_{11} = 0\) and (ii) is satisfied. The case when \(\displaystyle\limsup_{n \to \infty} S_n/ (2n \log n)^{1/2} = \infty\) a.s. is treated as well and it is shown how Theorem 2 by \textit{T. C. Hu} and \textit{N. C. Weber} [ibid. 45, No. 3, 479-482 (1992; Zbl 0739.60025)] should be modified. For an array of centered square integrable independent but in general nonidentically distributed r.v.'s sufficient conditions are given to guarantee that \(\displaystyle \limsup_{n \to \infty} S_n / (2\operatorname {var} S_n \log n)^{1/2} = 1 \text{a.s.} \) As an application one-sided logarithmic law is obtained for the bootstrapping mean.