I'm given that $\sqrt{n}(\hat \theta_n - \theta) \rightarrow_d W \sim N(0,\theta^2)$ and that $\hat \theta_n \rightarrow_p \theta$. I am trying to find an approximation for $P(|\hat \theta_n -\theta|>k)$ that does not depend on $\theta$. Presumably it depends on $n$.
I know that $(\hat \theta_n - \theta) \rightarrow_d Z \sim N(0, \frac{1}{n}\theta^2)$. Is it justified to say that for a fixed $n$, $(\hat \theta_n - \theta)$ is distributed similarly to $N(0,\frac{1}{n}\theta^2)$?
My strategy was to employ Markov's or Chebyshev's Inequality, but I cannot get rid of $\theta$. I am questioning if there can be any meaningful approximation at all.
I believe $|\theta_n-\theta|$ will have folded normal distribution. It's mean will depend on variance, which depends on $\theta$, so Markov does not work.
Any help will be very appreciated! Thanks!