In article <[EMAIL PROTECTED]>,
Thomas Souers <[EMAIL PROTECTED]> wrote:
> Hi, I am having some difficulties with the interpretation of particular modes of
>convergence. I was wondering if someone could clarify what convergence in
>distribution and in probability really mean in English (if this is possible).
> For example, with convergence in distribution I understand that although we say a
>sequence of random variables X1, ..., Xn converges in distribution to another random
>variable X (with cdf F(x)), it is really the cdfs F1, ..., Fn that converge to F(x).
>In the case of the arithmetic mean, for example, a sequence of random variables would
>consist of x-bar for sample sizes i=1,...,n. Each of these x-bars has a sampling
>distribution and hence a pdf and cdf. Does convergence in distribution then mean that
>as n
>increases towards infinity, x-bar will have of course a sampling distribution with
>pdf and cdf, and the cdf will closely approximate that of another random variable? I
>know that this is probably a stupid question that is common sensical to many people,
>but I am having some difficulties with the concept.
> Secondly, I would like to understand the English analogue (if it exists) of
>convergence in probability. For example, if X-bar converges in probability to mu as n
>approaches infinity then does this mean that although x-bar may not be close to mu
>for a particular sample, that over all possible samples it will be close to mu?
> Thank you for your help.
Something happens "in probability" if the probability that
it does not happen at time n goes to 0. For convergence,
what has to happen is that whatever amount of closeness is
desired, this amount happens in probability.
For "probability one", the change is that the probability
is one that it will happen from some point on.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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