Sums of random variables mod p

Question

Let $\varepsilon_1, \ldots, \varepsilon_n$ be independent random variables taking values $0,1$ each with probability $1/2$. It is well known that $R_n=\varepsilon_1+ \cdots+ \varepsilon_n$ modulo a prime $p$ tends to the uniform distribution on $\mathbb{Z}_p$ (say, in total variation distance, but also in a lot of other senses).

Is it known what the speed of convergance is? Let us say I want to bound $\delta_n=\sup_{k}|\mathbb{P}(\varepsilon_1+ \cdots+ \varepsilon_n=k)-1/p|$ in terms of $n$. Is it true that $\delta_n$ converges to $0$ exponentially? If this is true, could you provide a decent reference?

Many thanks for the attention.

Answer 1

Yes, and it's proportional to $\cos ( \pi/p)^n$. To see this, observe that adding one more independent random variable acts on the probability distribution as a linear operator, hence a $p \times p$ matrix. Using Fourier analysis mod $p$, compute the eigenvalues of this matrix. Note that one is $1$, corresponding to the uniform distribution, and the rest are at most $cos (\pi/p)$.

More generally, any Markov chain converges exponentially to a stable distribution.

Not a great reference but here's Wikipedia.