Limit distributions of random variables connected with long duplications in a sequence of 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 long duplications
$$ ((X_i,X_{i+1},\dots,X_{i+n-1})=(X_j,X_{j+1},\dots,X_{j+n-1}),\quad1\le i<j\le N), $$
for the distribution of the waiting time until the first duplication of a given length and for the distribution of the maximal duplication length.