Further going through old lecture notes I've stumbled upon this...
Let's say we are dealing with a sequence of random variables $\{X_n\}_{n=1}^\infty$ such that $\sqrt{n}(X_n-1)\to N(0,2)$ in distribution. Our aim is to find limiting distribution of standarized sequence $\{\sqrt{X}_n\}_{n=1}^\infty$.
to be quite honest I have no idea how to approach this task. I tried to go through other notes I have, but other than CLT and Slutsky's Theorem not much comes to my mind. I've tried to work with that from the definition, to try to somehow transform a sequence of interest into a more appealing form but then I am mostly stuck. First o all I don't know a thing about its Variance or mean. It seems as though, it would be some transformation of limiting distribution parameters from the original series, but again I have no clue how to approach this.
I'd be most grateful if someone could lend me a hand in this task.