Repeatedly rolling a die and the tails of the multinomial distribution. 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.

The variance of $X_i$ is $\frac{m(m+2)}{12}$ so of $Y$ is $\frac{m+2}{12mn}$. The central limit theorem leads to a normal approximation for moderate to high $n$  
$$\Pr\left( Y \geq k\right) \approx 1 - \Phi\left((2k-1){\sqrt{\frac{3mn}{m+2}}}\right)$$
Stein's method and other similar methods can help calculate the rates of convergence 
