Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables such that \(X_{n}\) has a \(\mathrm{N}\left(0, \sigma_{n}^{2}ight)\) distribution, conditional on \(\sigma_{n}\). For each case detailed below, determine if the sequence \(\left\{X_{n}ight\}_{n=1}^{\infty}\) is uniformly integrable and determine if \(X_{n}\) converges weakly to a random variable \(X\) as \(n ightarrow \infty\).

a. \(\sigma_{n}=n^{-1}\) for all \(n \in \mathbb{N}\).

b. \(\sigma_{n}=n\) for all \(n \in \mathbb{N}\).

c. \(\sigma_{n}=10+(-1)^{n}\) for all \(n \in \mathbb{N}\).

d. \(\left\{\sigma_{n}ight\}_{n=1}^{\infty}\) is a sequence of independent random variables where \(\sigma_{n}\) has an \(\operatorname{ExpOnential}(\theta)\) distribution for each \(n \in \mathbb{N}\).