Distribution of sample mean

Distribution of sample mean

Define a sequence of functions $f_i(x) = a^{x-t_i}$ for $t_{i+1} \geq x
\geq t_i$ and $0$ otherwise, where $0 \leq a\leq 1$. The values $t_i$ are
strictly increasing and $t_0 = 0$.
If you sample $k$ values $x_j$ uniformly at random from an interval
$[0,\dots,n]$, what is the distribution of $\sum_{j=1}^k f(x_j)/k$ as a
function of the values $t_i, k$ and $n$?