Erdős-Rényi law for stationary Gaussian sequences (Q803635)

The Erdős-Rényi strong law of large numbers for the increments of partial sums of i.i.d. r.v.s is extended to stationary Gaussian sequences. Let \(\{X_ j\}\) denote such a sequence with E \(X_ 1=0\), E \(X^ 2_ 1=1\). Set \(r_ n=E X_ 1X_{n+1}\), \(S_ n=\sum^{n}_{1}X_ j\), \(\sigma^ 2_ n=E S^ 2_ n\), and \[ C(n,k)=\max_{0\leq j\leq n-k}(S_{j+k}-S_ j)/(k\sigma^ 2_ k)^{1/2}. \] If either (i) \(r_ n\leq 0\) \((n=1,2,...)\) or (ii) \(r_ n=o(n^{-\nu})\) for some \(\nu >0\), then one has, for any \(c>0\), \[ (1)\quad \lim_{n\to \infty}C(n,[c \log n])=(2/c)^{1/2}\quad a.s. \] The result is also extended to other increments \(C(n,a_ n)\), where \(\{a_ n\}\) denotes an integer sequence with \(a_ n=o(n^{\alpha})\) for all \(\alpha >0\), replacing the special sequence \(a_ n=[c \log n]\) in assertion (1).