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Convergence in distribution

  1. Sep 24, 2008 #1
    Given the definition:
    For real-valued random variables [tex]X_n[/tex], [tex]n\geq1[/tex] and X, then [tex]X_n\stackrel{D}{\rightarrow}X[/tex] if for every bounded continuous function g: [tex]R \rightarrow R[/tex], [tex]E_n[/tex][g([tex]X_n[/tex])][tex]\rightarrow E[/tex][g(X)]

    I want to prove the continuous mapping theorem:
    If [tex]X_n\stackrel{D}{\rightarrow}X[/tex] then [tex]h(X_n)\stackrel{D}{\rightarrow}h(X)[/tex] for any continuous function h: [tex]R \rightarrow R[/tex]
    without using Skorokhod's representation theorem.

    The theorem makes sense to me intuitively but I'm lost as to how to prove it mathematically.

    Edit: apologies for the really bad latex, my browser keeps hanging on the preview/save
    Last edited: Sep 24, 2008
  2. jcsd
  3. Sep 28, 2008 #2
    If anyone was interested:

    Say [tex]h(Y_n) = Z_n, h(Y) = Z[/tex]

    [tex]E(g(Z_n)) \rightarrow E(g(Z))[/tex] for every g that is bounded and continuous (from definition)

    [tex]E(f(Y_n)) \rightarrow E(f(Y))[/tex] for every f that is bounded and continuous (from definition)

    [tex]E(g(h(Y_n)) \rightarrow E(g(h(Y))[/tex] is true because h is continuous and g o h is also continuous, h is also bounded by g
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