# Lecture 14. D) Slutsky’s Theorem

If a sequence $X_{n}$ converges in distribution to $X$, and a sequence $Y_{n}$ converges in probability to a constant $c$, then
• $X_{n}.Y_{n}\overset{d}{\rightarrow}c.X$
• $X_{n}+Y_{n}\overset{d}{\rightarrow}c+X$
• $\frac{X_{n}}{Y_{n}}\overset{d}{\rightarrow}\frac{X}{c}$ if $c\neq0$
The results above also holds if $X_{n}$ converges in probability, in which case the implications also apply to convergence in probability.