When can I change the order of plim and Lim?

  • Thread starter cuak2000
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  • #1
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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 realise using infinity here is rather sloppy, but I hope it doesn't cause confusion)

Thanks for your help!
 

Answers and Replies

  • #2
336
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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).
 
  • #3
8
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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.
 

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