Asymptotics of integer compositions

Question

A (weak) composition of a positive integer $n$ into $k$ parts is an ordered sequence of nonnegative integers $(a_1, a_2, \ldots, a_k)$ such that $ \sum_{i=1}^k a_i = n $. I am interested in the case when the parts are bounded: $a_i\in\{0, 1, \ldots, j-1\}$. The number of such compositions satisfies the two-variable recurrence \begin{equation} \kappa(n,j,k)=\sum_{i=0}^{j-1}\kappa(n-i,j,k-1) \end{equation} and can be expressed as \begin{equation} \kappa(n,j,k) = \sum_{s\geq0} (-1)^s {k \choose s} {k-sj+n-1 \choose k-1} \end{equation} [R. P. Stanley, Enumerative Combinatorics, Vol. I, p. 307]. Does someone know how to work out the asymptotics of $\kappa(n,j,k)$ when $k\to\infty$, $n\sim\lambda k$, $\lambda\in(0,j-1)$, and $j$ is fixed?