Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 31 / 2 / b / Solution

For an open neighborhood D of mu, its complement is closed and excludes mu. The preceding upper bound is minus infinity, so, for example, eventually $\mathbb{P}(X_n\notin D)\le e^{-b_n}\to 0$. Hence the sequence concentrates in every neighborhood of mu.

If an open E contains mu, its probability tends to one. Its logarithm divided by $b_n\to \infty$ therefore tends to zero, which equals $-{\inf }_E{I}^{\prime }$. If E does not contain mu, that latter value is minus infinity and its lower bound is automatic. ThusWith the closed-set upper bound this proves the slower-speed degeneration of a large deviation principle. This degenerate rate has no strictly positive finite value, so it does not give the sequence a different Hurstiness under the stated definition.