# Concentration inequality for mean of Gaussian mixture

Say I have i.i.d. samples $$X_1, \ldots, X_n \sim p \mathcal{N}(\mu_1, \sigma^2) + (1 - p) \mathcal{N}(\mu_2, \sigma^2)$$. Then suppose I estimate the mean with the sample mean

$$\widehat{\mu} = \frac{1}{n} \sum_{j = 1}^n X_j$$

Clearly $$\widehat{\mu}$$ is unbiased since $$\mathbb{E}[\widehat{\mu}] = p\mu_1 + (1 - p)\mu_2$$. How can I get a concentration inequality similar to a gaussian tail bound? Recall that if we had i.i.d. samples $$Y_1, \ldots, Y_n \sim \mathcal{N}(\mu, \sigma^2)$$ then we know

$$\mathbb{P}\left(\left\lvert\frac{1}{n}\sum_{j=1}^n Y_j - \mu\right\lvert > \varepsilon\right) \leq \exp\left(-\frac{n\varepsilon^2}{2\sigma^2}\right)$$

Can we get a similar style concentration inequality for $$\widehat{\mu}$$? That is, I'm looking for something like

$$\mathbb{P}\left(\left\lvert\widehat{\mu} - (p\mu_1 + (1 - p)\mu_2)\right\lvert > \varepsilon\right) \leq \: ???$$