  We study analytically the distribution of the minimum of a set of
hierarchically correlated random variables $E_1$, $E_2$, $...$, $E_N$ where
$E_i$ represents the energy of the $i$-th path of a directed polymer on a
Cayley tree. If the variables were uncorrelated, the minimum energy would have
an asymptotic Gumbel distribution. We show that due to the hierarchical
correlations, the forward tail of the distribution of the minimum energy
becomes highly nnon universal, depends explicitly on the distribution of the
bond energies $\epsilon$ and is generically different from the
super-exponential forward tail of the Gumbel distribution. The consequence of
these results to the persistence of hierarchically correlated random variables
is discussed and the persistence is also shown to be generically anomalous.
