Strong approximation of additive functionals (Q1200245)

Let \(X_ 1,X_ 2,\dots\) be a sequence of i.i.d. random variables and put \(S_ 0=0\), \(S_ i=X_ 1+X_ 2+\cdots+X_ i\), \(i=1,2,\dots\). The almost surely properties of \(A_ N=\sum^ N_{i=1} f(S_ i)\) are investigated, where \(f(x)\) is a real function. The asymptotic behaviour of \(A_ N\) depends upon the properties of \(X_ i\) and \(f\). There is a quite big list of references where in particular the weak convergence of suitably normalized and centred \(A_ N\) is investigated. The authors give strong approximation of \(A_ N\) with the help of suitably constructed standard Wiener process and its functions. As a result they prove analogous results mentioned above where the weak convergence is replaced by the convergence with probability one. In the same way the authors investigate the functional of the form \(\int^{\lambda t}_ 0 f(w(s))ds\), \(t>0\), as \(\lambda\to\infty\), where \(w(s)\) is a standard Wiener process. Explicit formulations of the results are rather complicated.