Relation between convergence in distribution and in probability

Question

Does convergence in distribution imply convergence in probability ?

I suppose no, but I need a counterexample. Does anyone know any counterexamples ?

Answer 1

Consider the probability space $((0,1),\mathcal{B}(0,1))$ endowed with the Lebesgue measure $\lambda$ and the random variables $$X(\omega) := 1_{(0,1/2)}(\omega) \qquad \qquad Y(\omega) := 1_{(1/2,1)}(\omega), \qquad \omega \in (0,1).$$ Then $X \sim Y$. Set $X_n(\omega) := Y(\omega)$ for all $n \in \mathbb{N}, \omega \in (0,1)$.

1. $X_n \to X$ in distribution since $X_n \sim X$ for any $n \in \mathbb{N}$
2. $X_n$ does not converge in probability to $X$ since $$\lambda(|X_n-X|>1/2)= \lambda(|X-Y|>1/2) = 1.$$