- #1
cuak2000
- 8
- 0
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 [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 realize using infinity here is rather sloppy, but I hope it doesn't cause confusion)
Thanks for your help!
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 realize using infinity here is rather sloppy, but I hope it doesn't cause confusion)
Thanks for your help!