# Continuous mapping theorem and random vectors

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,

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

• I wander why you used the symbols $\mu$ and $\sigma$, random vectors converge to random vectors, not to moments. – carlo Nov 11 '19 at 22:49
• 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"?? – whuber Nov 12 '19 at 15:07
• 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. – Marj Nov 12 '19 at 19:10
• en.wikipedia.org/wiki/Continuous_mapping_theorem#Statement – whuber Nov 13 '19 at 14:27