I am considering the following problem:

Let $F_n(\boldsymbol{\theta})$ is a sequence of random variables depending on a set of parameters $\boldsymbol{\theta}$ and the function mapping from $\boldsymbol{\theta}$ to $F_n$ is continuous. Now assume that $$F_n(\boldsymbol{\theta})\overset{P}{\to}F(\boldsymbol{\theta})$$ and $$\boldsymbol{\theta}_n\overset{P}{\to}\boldsymbol{\theta}\,$$.

Then would the following be correct? $$F_n(\boldsymbol{\theta}_n)\overset{P}{\to}F(\boldsymbol{\theta})\,$$.

If it is correct, could anyone provide a proof of this? Thanks!

  • $\begingroup$ You might want to add continuity conditions for the way $F_n$ depend on $\theta$. If no such assumption is made, the statement is clearly false. $\endgroup$ – Vim Sep 24 '17 at 1:52
  • $\begingroup$ @Vim thanks for mentioning this. Just added the condition, any idea how to prove the final statement? $\endgroup$ – user99015 Sep 24 '17 at 2:02
  • $\begingroup$ my idea would be to show $F_n(\theta_n)\to F_n(\theta)$ in probability. $\endgroup$ – Vim Sep 24 '17 at 2:39
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    $\begingroup$ This isn't even necessarily true if $\theta_n$ is deterministic! $\endgroup$ – Jason Sep 24 '17 at 3:55
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    $\begingroup$ Could you be a little more precise about what "$F_n(\theta)\to F(\theta)$ in probability" means? Are you regarding the functions $F_n$ to be random elements in some space of functions? If so, with what topology? $\endgroup$ – kimchi lover Sep 24 '17 at 12:10

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