  In this article, we consider scenarios in which traditional estimates for the
active subspace method based on probabilistic Poincar\'e inequalities are not
valid due to unbounded Poincar\'e constants. Consequently, we propose a
framework that allows to derive generalized estimates in the sense that it
enables to control the trade-off between the size of the Poincar\'e constant
and a weaker order of the final error bound. In particular, we investigate
independently exponentially distributed random variables in dimension two or
larger and give explicit expressions for corresponding Poincar\'e constants
showing their dependence on the dimension of the problem. Finally, we suggest
possibilities for future work that aim for extending the class of distributions
applicable to the active subspace method as we regard this as an opportunity to
enlarge its usability.
