Random Forests are widely claimed to capture interactions well. However, some
simple examples suggest that they perform poorly in the presence of certain
pure interactions that the conventional CART criterion struggles to capture
during tree construction. We argue that alternative partitioning schemes can
enhance identification of these interactions. Furthermore, we extend recent
theory of Random Forests based on the notion of impurity decrease by
considering probabilistic impurity decrease conditions. Within this framework,
consistency of a new algorithm coined 'Random Split Random Forest' tailored to
address function classes involving pure interactions is established. In a
simulation study, we validate that the modifications considered enhance the
model's fitting ability in scenarios where pure interactions play a crucial
role