Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) and \(\left\{Y_{n}ight\}_{n=1}^{\infty}\) be sequences of random variables that converge in distribution as \(n ightarrow \infty\) to the random variables \(X\) and \(Y\), respectively. Suppose that \(X_{n}\) has a \(\mathrm{N}\left(0,1+n^{-1}ight)\) distribution for all \(n \in \mathbb{N}\) and that \(Y_{n}\) has a \(\mathrm{N}\left(0,1+n^{-1}ight)\) distribution for all \(n \in \mathbb{N}\).

a. Identify the distributions of the random variables \(X\) and \(Y\).

b. Find some conditions under which we can conclude that \(X_{n} Y_{n}^{-1} \xrightarrow{d} Z\) as \(n ightarrow \infty\) where \(Z\) has a \(\operatorname{CAUChy}(0,1)\) distribution.

c. Find some conditions under which we can conclude that \(X_{n} Y_{n}^{-1} \xrightarrow{d} W\) as \(n ightarrow \infty\) where \(W\) is a degenerate distribution at one.