My question is on whether or not continuous mapping theorem can be applied to elements of a random vector.

Consider $[X_n,Y_n] \rightarrow [\mu, \sigma]$

Would it also be true that for any continuous function $f$ on all real numbers,

$f[X_n,Y_n] \rightarrow f[\mu,\sigma]$

  • $\begingroup$ I wander why you used the symbols $\mu$ and $\sigma$, random vectors converge to random vectors, not to moments. $\endgroup$ – carlo Nov 11 '19 at 22:49
  • $\begingroup$ Could you please explain what the notation "$f\left[X_n,Y_n\right]$" might possibly mean when $f$ is a function "on all real numbers"?? $\endgroup$ – whuber Nov 12 '19 at 15:07
  • $\begingroup$ I mean like f(X,Y)=X+Y or f(X,Y)=log(X/Y). To give more context, say sample mean converges to population mean. By continuous mapping, f(sample mean) converges to f(pop mean), right? So what I am asking is if instead of sample mean, I have vector of 2 sample means. $\endgroup$ – Marj Nov 12 '19 at 19:10
  • $\begingroup$ en.wikipedia.org/wiki/Continuous_mapping_theorem#Statement $\endgroup$ – whuber Nov 13 '19 at 14:27

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