 Solution

For fixed $k$, independence and the product large-deviation principle give the vector $(Y_n(1),\dots ,Y_n(k))$ the good rate function ${\sum }_{j=1}^{k}I(y_j)$. Since the minimum is a continuous map, the contraction principle for large deviations givesIn every vector in this fibre, at least one coordinate equals $m$ and all others are at least $m$. The large-deviation rate of a minimum of independent copies therefore reduces the infimum toIf $0<m<\mu ={\lambda }^{-1}$, the other coordinates can equal $\mu$ and cost zero. If $m\ge \mu$, the rate function is increasing on $[\mu ,\infty )$, so every coordinate costs at least $I(m)$ and equality is achieved when all coordinates equal $m$. For $m\le 0$ the cost is infinite. ThusAt the threshold both expressions are zero. Goodness follows from the contraction principle for large deviations. A small minimum requires only one atypically small mean; a large minimum requires all $k$ means to be atypically large, which explains the two costs.