On the asymptotic of lottery numbers

Let $L(n,k,r,p)$ denote the minimum number of $k$-subsets of an $n$-set such that all the $\binom{n}{p}$ $p$-subsets are intersected by one of them in at least $r$ elements. The case $p=r$ corresponds to the covering numbers, while the case $k=r$ corresponds to the Tur\'an numbers. In both cases, there exists a limit of $L(n,k,r,p) / \binom{n}{r}$ as $n\to\infty$. We prove the existence of this limit in the general case.