$X,Y$ have the same distribution, then $X_n\stackrel{D}{\rightarrow}X$, then $X_n\stackrel{D}{\rightarrow}Y$

I am trying to understand convergence in law. My question is the following:

Given two random variables $X,Y$ with the same distribution, is it correct to say that if a sequence of random variables $X_n$ converges in law to $X$, i.e. $X_{n}\stackrel{D}{\rightarrow}X$, it also converges in law to $Y$ since it has the same distribution, i.e. $X_{n}\stackrel{D}{\rightarrow}Y$ ?

• Yes, because - as you wrote - $X$ and $Y$ have the same distribution (AKA law). – Fnacool May 29 '17 at 18:12
• Things become clear if you interpret $X_{n}\stackrel{D}{\rightarrow}X$ in words by: "the distribution of $X_n$ converges to the distribution of $X$". Then shows directly that it is legal to replace $X$ by $Y$ in this context. – drhab May 29 '17 at 19:05

Let $F(x)$ be the c.d.f. of both $X$ and $Y$, as they have the same distribution.
Let $F_n(x)$ be the c.d.f. of $X_n$, for each $n\in\mathbb{N}$.
By definition, $X_{n}\stackrel{D}{\rightarrow}X$ means that $\lim\limits_{n\to\infty}F_n(x) = F(x),$ for all $x\in\mathbb{R}$ (at which $F$ is continuous), thus it becomes obvious that $X_{n}\stackrel{D}{\rightarrow}Y$, as $F(x)$ is the c.d.f. of $Y$.