An Erd\" os-R\' enyi law for nonconventional sums

We obtain rge Erd" os-R' enyi type law of large numbers for “nonconventional” sums of the form $S_n=\sum^n_{m=1}F(X_m,X_{2m},…,X_{\ell m})$where$X_1,X_2,…$isasequenceofi\mathrm{.}i\mathrm{.}d\mathrm{.}randomvariablesand$F$ is a bounded Borel function. The proof relies on nonconventional large deviations obtained in [8].