It is well known that for a set $A$ of integers, if $gcd(A) = d$,
then the (integer) linear combination of $A$ is $d\mathbb{Z}$.
I'm looking for a probability generalization of this, namely the following.

Let $\varepsilon>0$, a finite set $A$ of positive integers with $gcd(A) = d$.
Let $N$ be   large (depending on $A,\varepsilon$) and $\alpha\in A^N$ such that,
the density of every  $a\in A$ in $\alpha$ satisfies
$|\alpha^{-1}(a)|/N\geq \varepsilon$.
Let  $CCS(\alpha)$ (consecutive sum set) denote the set of $b\in\mathbb{N}$ such that for some $n,m$, $b = \alpha(n)+\alpha(n+1)+\cdots+\alpha(n+m-1)$.

**Question**: Do we have, 
 $|CCS(\alpha) |/\sum_n\alpha(n)\geq (1-\varepsilon)/d$.