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I am reading Kai Lai Chung's A Course in Probability Theory. I understood the concept of vague convergence of a sequence of probability measures in the book. But it seems to me the book uses the concept of vague convergence of a sequence of distribution functions without defining it first. But I could miss the definition, and if you can point me where the book defines it, it will be nice.

So I was wondering

  1. how the vague convergence of a sequence of distribution functions is defined?
  2. how it is related to the vague convergence of the corresponding probability measures?
  3. how it is related to the point-wise convergence of the distribution functions?
  4. ADDED: for a sequence of (sub)probability measures or a sequence of distribution functions, how weak convergence is defined? How it and vague convergence are different from each other?

Thanks and regards!

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    It is defined, on p. 89: "We say that $F_n$ converges vaguely to $F$ and write..."2011-03-16
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    @Shai: Thanks! That answered my questions 1 and 2. Do you know for a sequence of (sub)probability measures or a sequence of distribution functions, how weak convergence is defined and how it and vague convergence are different from each other?2011-03-16
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    As for 4), weak convergence is the same as convergence in distribution. That is, a sequence of distribution functions $F_n$ converges to $F$ weakly if and only if $F_n (x) \to F(x)$ for any continuity point $x$ of $F$.2011-03-16
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    For the relation between different types of convergence of distribution functions, see pp. 9-10 in http://neptune.galaxy.gmu.edu/stats/syllabi/it971/Lecture7.pdf (if $F$ is proper then the definitions are equivalent).2011-03-16

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