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Can anyone explain the syntax from the following Wikipedia article on the convergence of random variables?

alt text

In particular:

  1. What is the squiggly line?
  2. What are $d$, $D$, $L$, $L_X$?
  3. Why does it say $X_n$ implies X? They don't seem to be statements
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From the article: "Convergence in distribution may be denoted as [all this variants]". What is your question? – Grigory M Aug 7 '10 at 10:31
@Grigory M: Updated the question – Casebash Aug 7 '10 at 10:52
All these symbols are just variants of notation for convergence in distribution. You don't ask "what are these c, o and s you need to multiply to get $\cos$", right? – Grigory M Aug 7 '10 at 11:19
@Grigory: I really do need to read more carefully – Casebash Aug 7 '10 at 12:19
up vote 4 down vote accepted

All of these expressions are just notations for "X_n converges to $X$ in distribution".

  1. The squiggly line is really just a notation (a rather strange one) for convergence in distribution.

  2. The symbols $d$ and $\mathcal D$ stand for distribution, $\mathcal L$ and $\mathcal L_X$ stand for law. (One also speaks of "convergence in law" instead of "convergence in distribution".

  3. This is also just a notation for convergence in distribution (also a rather strange one); the arrow does not symbolize an implication of statements.

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Oh, I'm feeling quite thick at the moment – Casebash Aug 7 '10 at 11:40

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