On the asymptotic behavior of a sequence of random variables of interest in the classical occupancy problem

In the classical occupancy problem one puts balls in $n$ boxes, and each ball is independently assigned to any fixed box with probability $\frac {1}{n}$. It is well known that, if we consider the random number $T_n$ of balls required to have all the $n$ boxes filled with at least one ball, the sequence $\{T_n/(n\log n)\colon n\geq 2\}$ converges to 1 in probability. Here we present the large deviation principle associated to this convergence. We also discuss the use of the Gärtner Ellis Theorem for the proof of some parts of this large deviation principle.