Let (X_{n}, Y_{n}: Omega ightarrow mathbb{R}, n geqslant 1), be two sequences of random variables. a) If

Question:

Let $X_{n}, Y_{n}: \Omega ightarrow \mathbb{R}, n \geqslant 1$, be two sequences of random variables.

a) If $X_{n} \xrightarrow{d} X$ and $Y_{n} \xrightarrow{\mathbb{P}} c$, then $X_{n} Y_{n} \xrightarrow{d} c X$. Is this still true if $Y_{n} \xrightarrow{d} c$ ?

b) If $X_{n} \xrightarrow{d} X$ and $Y_{n} \xrightarrow{\mathbb{P}} 0$, then $X_{n}+Y_{n} \xrightarrow{d} X$. Is this still true if $Y_{n} \xrightarrow{d} 0$ ?

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