A functional limit theorem for Erdős and Rényi's law of large numbers
We use the theory of large deviations on function spaces to extend Erdős and Rényi's law of large numbers. In particular, we show that with probability 1, the double-indexed set of paths \(\{W_{N,n}\}\) defined by \[ W_{N,n} (t,\omega)={S_{n+[h(N)t]} (\omega)-S_ n (\omega) \over h(N)}+{h(N)t-[h(N)t] \over h(N)} X_{n+[h(N)t]+1} (\omega), \] where \(S_ n={1 \over n} \sum^ n_ 1X_ i\), \(\{X_ i:i \geq 1\}\) is an i.i.d. sequence of random variables, and \(h(N)=[c \log N]\) is relatively compact; the limit set is given by the set \([x:I^*(x) \leq 1/c]\) where \(I^*(x)=\int^ 1_ 0 I(x'(t)) dt\) and \(I\) is Cramér's rate function.
scientific article; zbMATH DE number 569019