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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 [itex] X_n(k) [/itex] of i.i.d. random variables that are a function of some variable k. In what case is

[itex]

plim_{n \rightarrow \infty }(lim_{k \rightarrow \infty} X_n(k) ) = lim_{k \rightarrow \infty }(plim_{n \rightarrow \infty} X_n(k) ) [/itex]

2) The particular one is: if I know that

[itex] lim_{k \rightarrow \infty }(plim_{n \rightarrow \infty} X_n(k) ) = \infty [/itex]

can I assure also that

[itex] plim_{n \rightarrow \infty }(lim_{k \rightarrow \infty} X_n(k) ) = \infty [/itex] ??

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

Thanks for your help!