Suppose real value random variables satisfy $X_{n} \Rightarrow X$ (convergence in distribution) as $n\to \infty$ in the same probability space $(\Omega, \mathcal F, \mathbb P)$. It is well known that $\lim_{n\to \infty} \mathbb E f(X_{n}) = \mathbb E f(X)$ for all continuous bounded real functions $f:\mathbb R \to \mathbb R$.

[Q.] If $f$ is continuous and linear growth, i.e. $|f(x)| < K(1 + |x|)$ for some constant $K$, can you find counter-example for $\lim_{n\to \infty} \mathbb E f(X_{n}) = \mathbb E f(X)$? What additional conditions are needed to still have $\lim_{n\to \infty} \mathbb E f(X_{n}) = \mathbb E f(X)$?