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.
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