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When can I change the order of plim and Lim?

  1. Oct 1, 2011 #1
    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

    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!
  2. jcsd
  3. Oct 3, 2011 #2
    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).
  4. Oct 6, 2011 #3
    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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