If $Z_n\overset{d}{\rightarrow}Z,\ Y_n\overset{d}{\rightarrow}a$ where $a\in\mathbb{R}$ where the CDF of Z is continuous then $Y_nZ_n\overset{d}{\rightarrow}aZ$

I tried proving using a similliar proof to Slutsky's theorem with sum and got that if I prove $$ \mathbb{P}(|Z_n(Y_n-a)|>\delta)\rightarrow 0$$
I finish.

I know that $Y_n \overset{\mathbb{P}}{\rightarrow}a$ So I feel like I am close to the answer, I havn't used the fact that the CDF of Z is continuous yet, so I feel like I'm supposed to use it here.

Is there a way to complete the proof?
If not, can I get a hint how to solve the problem and how to use the fact that the CDF of Z is continuous.

Thank you,

  • $\begingroup$ The hypothesis that $Z$ have no point masses is not needed. $\endgroup$
    – Andrew
    Jun 27 at 15:35
  • $\begingroup$ Yes, many thanks $\endgroup$ Jun 27 at 16:06
  • $\begingroup$ Now that I tried to prove it, I got stuck at showing that $|๐‘‹๐‘›(๐‘Œ๐‘›โˆ’๐‘)|\rightarrow 0$ weak convergence. The problem presented in the link needs probability convergence, but I wonder if its possible to prove weak convergence. $\endgroup$ Jun 27 at 16:52


Browse other questions tagged .