Suppose $\{X_n\}_{n\in\mathbb{N}}$ is a sequence of iid random variables with $X_n \sim \text{Beta}(1/2, 1)$. I would like to compute or get a good upper bound on$$ T = \lim_{d \to \infty} \frac{P\left(\displaystyle\sum_{i=1}^{d-K} X_i < Cd\right)}{P\left(\displaystyle\sum_{i=1}^d X_i < Cd\right)}$$where $K$ and $C$ are constant and $C < \mu_X$. I would like to apply the central limit theorem, but this doesn't work since $Cd$ varies with $d$. Another thing to try would be Berry-Esseen theorem, but this gives an $\mathcal{O}(1/\sqrt{d})$ error, which is insufficient given how quickly the numerator and denominator decay. What are some other approaches to this problem?
Note: An older version of this question was significantly different, so many of the comments here are outdated.
After thinking about the problem for longer, one option is to use Cramer's theorem to upper bound the numerator and lower bound the denominator, thus giving an upper bound for $T$. However, this requires me to compute the Legendre transform of $\log M(t)$ where $M(t)$ is the MGF of $\text{Beta}(1/2, 1)$. I've posted another question that addresses this subproblem here.