Lecture 14. D) Slutsky’s Theorem

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Slutsky’s Theorem

Slutsky’s Theorem provides some nice results that apply to convergence in distribution:

If a sequence [math]X_{n}[/math] converges in distribution to [math]X[/math], and a sequence [math]Y_{n}[/math] converges in probability to a constant [math]c[/math], then

  • [math]X_{n}.Y_{n}\overset{d}{\rightarrow}c.X[/math]
  • [math]X_{n}+Y_{n}\overset{d}{\rightarrow}c+X[/math]
  • [math]\frac{X_{n}}{Y_{n}}\overset{d}{\rightarrow}\frac{X}{c}[/math] if [math]c\neq0[/math]

The results above also holds if [math]X_{n}[/math] converges in probability, in which case the implications also apply to convergence in probability.