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Let $X_n$ and $Y_n$ converge in law to $X$ and $y$ respectively, where $y$ is a constant. Then are the following true?

  1. $X_nY_n \to yX$
  2. $X_n + Y_n \to X + y$
  3. $X_n/Y_n \to X/y$ (if $y\neq 0$)

These statements are obviously true for almost sure convergence, but what about convergence in probability? For convergence in distribution? In the convergence in distribution case I think we may have to suppose $Y_n$ converges to a constant.

I'd also be interested in references for these statements, because I can't find anything online.

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Firstly $Y_n \overset{d}\to y \implies Y_n \overset{P}\to y$, see here. So, you may directly apply Slutsky's theorem to obtain that all statements 1.-3. are indeed true under your assumptions. So, Slutsky's theorem and the properties listed in the link, answer indeed your questions:

  1. These statements are obviously true for almost sure convergence, but what about convergence in probability?: Yes, they are true (Note 3. in the link).
  2. For convergence in distribution? Yes, they are true (Note 1. in the link).
  3. In the convergence in distribution case I think we may have to suppose $Y_n$ converges to a constant. I do not directly understand this question, because you have already assumed that. But, yes, this assumption is necessary (Note 2. in the link).
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