Limit distributions of random variables connected with multiple long duplications in a sequence lof independent trials

Let $X_0,X_1,\dots$ be a sequence of independent trials with $m$ outcomes. We prove limit theorems for the distribution of the number of multiple long duplications
\begin{gather*} ((X_{i_1}, X_{i_1+1},\dots,X_{i_1+n-1})=(X_{i_t}, X_{i_t+1},\dots,X_{i_t+n-1}) \\ t=2,\dots,k,\quad1\le i_1<\dots<i_k\le N), \end{gather*}
for the distribution of the waiting time until the first multiple duplication of a given length and for the distribution of the maximal duplication length.