  In this article, we consider limit theorems for some weighted type random
sums (or discrete rough integrals). We introduce a general transfer principle
from limit theorems for unweighted sums to limit theorems for weighted sums via
rough path techniques. As a by-product, we provide a natural explanation of the
various new asymptotic behaviors in contrast with the classical unweighted
random sum case. We apply our principle to derive some weighted type
Breuer-Major theorems, which generalize previous results to random sums that do
not have to be in a finite sum of chaos. In this context, a Breuer-Major type
criterion in notion of Hermite rank is obtained. We also consider some
applications to realized power variations and to Ito's formulas in law. In the
end, we study the asymptotic behavior of weighted quadratic variations for some
multi-dimensional Gaussian processes.
