For $1\leq i\leq n$ let $X_i$ be independent random variables, and let each $X_i$ be the uniform distribution on the set ${0,1,2,\dots,m}$ so that $X_i$ is like an $m+1$ sided die. Let $Y=\frac{1}{n}\sum_{i=1}^n \frac{1}{m} X_i,$ so that $\mathbb{E}(Y)=\frac{1}{2}$. I am interested in the tails of this distribution, that is the size of $\Pr\left( Y \geq k\right)$ where $\frac{1}{2}< k\leq 1$ is a constant.
In the case where $m=1$, we are looking at the binomial distribution, and $\Pr\left( Y \geq k\right)= \frac{1}{2^n}\sum_{i=0}^{(1-k) n} \binom{n}{i}$ and we can bound this above by $(1-k)n \binom{n}{(1-k)n}$ and below by $\binom{n}{(1-k)n}$ which yields $\Pr\left( Y \geq k\right)\approx \frac{1}{2^n} e^{n H(k)}$ where $H(x)=-\left(x\log x+(1-x)\log (1-x)\right)$ is the entropy function. (I use approx liberally)
What kind of similar bounds do we have on the tails of this distribution when $m\geq 2$? I am looking to use the explicit multinomial properties to get something stronger than what you would get using Chernoff of Hoeffding.