When can I change the order of plim and Lim?

[cuak2000]

Hi. I have two questions, one general and one particular.

1) The general one is if you know or can give me a reference of when can I change the order of a probability limit and the pointwise limit of a function (assuming both plim and lim exist)
Say, take a sequence $X_n(k)$ of i.i.d. random variables that are a function of some variable k. In what case is

$plim_{n \rightarrow \infty }(lim_{k \rightarrow \infty} X_n(k) ) = lim_{k \rightarrow \infty }(plim_{n \rightarrow \infty} X_n(k) )$

2) The particular one is: if I know that

$lim_{k \rightarrow \infty }(plim_{n \rightarrow \infty} X_n(k) ) = \infty$

can I assure also that

$plim_{n \rightarrow \infty }(lim_{k \rightarrow \infty} X_n(k) ) = \infty$ ??


(i realize using infinity here is rather sloppy, but I hope it doesn't cause confusion)

Thanks for your help!

 

[Eynstone]

We must check whether all the limits exist in the first place.Although the general question is difficult to answer,I have a counterexample for the second ( where the second limit doesn't exist).

 

[cuak2000]

Thanks for your reply Eynstone.
Now I think the general problem is that lim[plim(X)]] makes sense (if both limits
exist at least), but that plim[lim(X)] maybe doesn't, since I'm not sure you can take a lim of a random variable.