Tight tail bounds for sums of random variables

Question

Let $X_1, X_2, \dots$ be iid uniformly on $[0,1]$. Define $Z_i^{(a)} = (X_i - a)^2$. Let $Y_n = \sum_{k=1}^n Z_k^{(1/k)}$. I am interested in matching tail bounds for $Y_n$ as $n \to \infty$. In particular, I would like to show that $$C_1e^{-C_2n} \leq P(Y_n < \theta n) \le C_3e^{-C_2n},$$ where $0< \theta < \mathbb{E}[Y_n]/n$ and $C_1, C_2, C_3$ depend only on $\theta$. The difficulty comes from the fact that both bounds have the same constant factor $C_2$ in the exponent. It would also be great to get an expression for $C_2$ in terms of $\theta$.