[proofplan]
The proof chains two previously established results. First, [Hoeffding's Lemma](/theorems/1956) promotes the bounded-range condition on each $W_i$ to a sub-Gaussian parameter $\sigma_i = (b_i - a_i)/2$. Second, [Sub-Gaussian Stability Under Linear Combinations](/theorems/1960) applied with weights $\gamma_i = 1/n$ shows that the sample average inherits the sub-Gaussian property with a computable parameter. Finally, the [Sub-Gaussian Tail Bound](/theorems/1953) converts the MGF condition into the desired exponential tail inequality.
[/proofplan]
[step:Promote each $W_i$ to sub-Gaussian via Hoeffding's Lemma]
For each $i \in \{1, \ldots, n\}$, the random variable $W_i$ satisfies $a_i \leq W_i \leq b_i$ almost surely. By [Hoeffding's Lemma](/theorems/1956), any random variable supported on an interval $[a_i, b_i]$ is sub-Gaussian with parameter $\sigma_i = (b_i - a_i)/2$. The sole hypothesis of Hoeffding's Lemma -- that $W_i$ takes values in a bounded interval almost surely -- is exactly the given assumption. Therefore each $W_i$ is sub-Gaussian with parameter
\begin{align*}
\sigma_i = \frac{b_i - a_i}{2}.
\end{align*}
[/step]
[step:Apply the stability theorem to the sample average with weights $\gamma_i = 1/n$]
The sample average $\bar{W} := \frac{1}{n}\sum_{i=1}^n W_i$ is the linear combination $\gamma^\top W$ with $\gamma_i = 1/n$ for each $i$. Since $W_1, \ldots, W_n$ are independent and each $W_i$ is sub-Gaussian with parameter $\sigma_i = (b_i - a_i)/2$, the hypotheses of [Sub-Gaussian Stability Under Linear Combinations](/theorems/1960) are satisfied. Applying that result, $\bar{W}$ is sub-Gaussian with parameter
\begin{align*}
\tilde{\sigma} = \left(\sum_{i=1}^n \gamma_i^2 \sigma_i^2\right)^{1/2} = \left(\sum_{i=1}^n \frac{1}{n^2} \cdot \frac{(b_i - a_i)^2}{4}\right)^{1/2} = \frac{1}{2n}\left(\sum_{i=1}^n (b_i - a_i)^2\right)^{1/2}.
\end{align*}
[guided]
Why these specific weights? The theorem concerns $\frac{1}{n}\sum_{i=1}^n W_i$, which is a linear combination of $W_1, \ldots, W_n$ with equal coefficients $\gamma_i = 1/n$. The stability theorem tells us that for independent sub-Gaussians, the sub-Gaussian parameter of a linear combination $\sum_i \gamma_i W_i$ is $(\sum_i \gamma_i^2 \sigma_i^2)^{1/2}$. We verify the hypotheses: independence is given, and each $W_i$ is sub-Gaussian with parameter $\sigma_i = (b_i - a_i)/2$ by the previous step.
Substituting $\gamma_i = 1/n$ and $\sigma_i = (b_i - a_i)/2$:
\begin{align*}
\tilde{\sigma}^2 = \sum_{i=1}^n \frac{1}{n^2} \cdot \frac{(b_i - a_i)^2}{4} = \frac{1}{4n^2}\sum_{i=1}^n (b_i - a_i)^2,
\end{align*}
so $\tilde{\sigma} = \frac{1}{2n}\left(\sum_{i=1}^n (b_i - a_i)^2\right)^{1/2}$.
[/guided]
[/step]
[step:Apply the sub-Gaussian tail bound to obtain the exponential inequality]
Since $\bar{W}$ is sub-Gaussian with parameter $\tilde{\sigma}$, the [Sub-Gaussian Tail Bound](/theorems/1953) gives, for all $t \geq 0$:
\begin{align*}
\mathbb{P}(\bar{W} - \mathbb{E}[\bar{W}] \geq t) \leq \exp\!\left(-\frac{t^2}{2\tilde{\sigma}^2}\right).
\end{align*}
Since $\bar{W} - \mathbb{E}[\bar{W}] = \frac{1}{n}\sum_{i=1}^n(W_i - \mathbb{E}[W_i])$, substituting $\tilde{\sigma}^2 = \frac{1}{4n^2}\sum_{i=1}^n (b_i - a_i)^2$:
\begin{align*}
\mathbb{P}\!\left(\frac{1}{n}\sum_{i=1}^n (W_i - \mathbb{E}[W_i]) \geq t\right) \leq \exp\!\left(-\frac{t^2}{2 \cdot \frac{1}{4n^2}\sum_{i=1}^n (b_i - a_i)^2}\right) = \exp\!\left(-\frac{2n^2 t^2}{\sum_{i=1}^n (b_i - a_i)^2}\right),
\end{align*}
which is the stated Hoeffding's inequality.
[/step]
