Let $(p_i)_i$ be i.i.d. Bernoulli variables with $P(p_i = 1) = 1/d$ and $P(p_i = 0) = 1 - 1/d$, where $d \geq 2$ is a fixed integer. Then $E(p_i) = 1/d$ and therefore by the strong law of large numbers,
$$ \overline{p}_n = \frac{p_0 + \cdots + p_{n-1}}{n} \to \frac{1}{d} $$
almost surely as $n \to \infty$. In particular, this implies that for each fixed $\varepsilon \in (0, 1/d)$,
$$ P\Big(\overline{p}_n < \frac{1}{d} - \varepsilon\Big) \to 0 $$
as $n \to \infty$. Moreover, the convergence should be exponential, in the sense that there exists a positive $\eta = \eta(\varepsilon) > 0$ such that
$$ P\Big(\overline{p}_n < \frac{1}{d} - \varepsilon\Big) = \mathcal{O}(e^{-\eta n}) $$
My question is: What is $\eta$ in terms of $\varepsilon$ and $d$? Can we compute the following?
$$ \eta(\varepsilon) = \lim_{n\to\infty} -\frac{1}{n}\log P\Big(\overline{p_n} < \frac{1}{d} - \varepsilon\Big) $$
In the case that $d = 2$, this has the interpretation of summing binomial coefficients in the sense that
$$ \sum_{k=0}^{\alpha n} \binom{n}{k} \leq 2^{H(\alpha)n} $$
where $\alpha \in (0,1/2)$ and $H(\alpha) = -\alpha \log_2 \alpha - (1-\alpha) \log_2(1-\alpha)$ is the binary entropy function. This would imply in this case that $\eta(\varepsilon) \leq H(1/2 - \varepsilon)$.
I wondered if there existed a similar bound for summing multinomial coefficients, though I am not familiar enough with probability theory to know how to compute this above limit (or how to prove the bound in the $d = 2$ case).
