Strassen's invariance principle for random subsequences
Let \(X_ n\), \(n\geq 1\), be independent r.v.'s such that \(EX_ n=0\), \(0<EX^ 2_ n < \infty\), and let \(S_ 0=0\), \(S_ n=X_ 1+\dots + X_ n\), \(n\geq 1\), \(ES^ 2_ n\to \infty\) as \(n\to \infty\). Denote by \(Z\) the random function obtained by the linear interpolation of \(S_ n\) at \(t_ n=ES^ 2_ n\) and put \[ Z_ n(t)=(2t_ n\log\log t_ n)^{-1/2}Z(t\cdot t_ n). \] The main theorem of the paper states that if \(\{N_ n,\;n\geq 1\}\) is a sequence of positive integer-valued r.v.'s satisfying \(t_{N_ n}/t_{k_ n}\to 1\) a.s., where \(\{k_ n,\;n\geq 1\}\) is a sequence of positive integers with \(k_ n\to \infty\) as \(n\to \infty\), then under some mild additional restriction on \(t_ n\), the net \(\{Z_{N_ n},\;n\geq 1\}\) is relatively norm-compact in \(C(\langle 0,1\rangle )\). Moreover, the set of its limit points coincides with Strassen's set \(K\) iff, for every \(\varepsilon > 0\), \(\sum^ \infty_{n=1}(\log t^*_{k_ n})^{\varepsilon-1}=\infty\), where \(t^*_{n+1}=\inf\{t_ m: t_ m\geq t^*_ n\exp(c/\log n)\}\), \(t^*_ 1=t_ 1\), \(t^*_ 2=t_ 2\), \(c>0\). The obtained result constitutes an extension of Theorems 2 and 3 by \textit{I. S. Chang} and \textit{C. A. Hsiung} [Z. Wahrscheinlichkeitstheorie Verw. Gebiete 64, 401- 409 (1983; Zbl 0506.60028)]. A similar result for a sequence of zero-mean square-integrable martingale differences is also given.
scientific article; zbMATH DE number 35547
scientific article