Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables such that \(\sigma_{n}^{-1}\left(X_{n}-\muight) \xrightarrow{d} Z\) as \(n ightarrow \infty\) where \(Z\) is a \(\mathrm{N}(0,1)\) random variable and \(\left\{\sigma_{n}ight\}_{n=1}^{\infty}\) is a real sequence such that

\[\lim _{n ightarrow \infty} \sigma_{n}=0\]

Consider the transformation \(g(x)=a x+b\) where \(a\) and \(b\) are known real constants, and \(a eq 0\). Derive the asymptotic behavior, normal or otherwise, of \(g\left(X_{n}ight)\) as \(n ightarrow \infty\).