The central limit theorem for the number of partial long duplications

Let $X_1,X_2,\dots$ be a sequence of independent random variables, $X_i=1,2,\dots$. We prove the central limit theorem for the number of sets ($i_1,\dots,i_m$), $1\le i_1<\dots<i_m\le n$, such that the conditions
$$ X_{i_1+k}=\dots=X_{i_m+k} $$
are satisfied for exactly $s-d$ values of $k\in\{0,\dots,s-1\}$.