Suppose we are performing a random walk in a group. More precisely, we have a finite generating set $S$ of a group $G$ and the probability of walking along generator $s$ is given by $\mu(s)$ for some fixed probability measure $\mu$ on $S$. Denote by $W(n, \mu)$ a random walk of length $n$ with respect to $\mu$, and let $A \subset G$. I can show the following implication. ($\mu_{unif}$ is the uniform probability measure.)

There is a $c < 1$ such that $P(W(n, \mu_{unif}) \in A) < c^n$ for sufficiently large $n$ $$\Rightarrow$$ For any probability measure $\mu$ supported on $S$, there is a $d < 1$ such that $P(W(n, \mu) \in A) < d^n$ for sufficiently large $n$

I can fairly easily prove this (in about a page) with only elementary arguments, so I am guessing this is known. Is there a reference with this statement somewhere? Perhaps, there is a one-line proof using some other result?