Skorokhod Representation of a Random Variable

See 3.12 of Williams.

Let X^{-}(\omega) = \inf \{ x \mid F(z) \geq \omega \} where F : \mathbb{R} \rightarrow [0,1] and

  1. x \leq y \implies F(x) \leq F(y).
  2. \lim_{x \rightarrow \infty} F(x)= 1 and \lim_{x \rightarrow -\infty} F(x) = 0.
  3. F is right continuous.

Then by the proof X^{-}(\omega) \leq c \implies \omega \leq F(c). Thus c \not \in \{ y \mid F(y) < \omega \} and is therefore an upper bound (if not then \exists y such that F(y) < \omega and c < y but by monotonicity F(c) \leq F(y)). Therefore c \geq \sup \{ y \mid F(y) < \omega \}.

On the other hand suppose c < \inf \{ x \mid F(x) \geq \omega \} then F(c) < \omega (if not then F(c)  \geq \omega but then \inf \{ x \mid F(x) \geq \omega \} is a lower bound for all such c which would imply \inf \{ x \mid F(x) \geq \omega \} \leq c). Now \sup \{ y \mid F(y) < \omega \} is an upper bound for any x such that F(c) < \omega so c \leq \sup \{ y \mid F(y) < \omega \}. Now suppose c = \sup \{ y \mid F(y) < \omega \} then F(c + 1/n) \geq \omega and by right continuity this implies F(c) \geq \omega. Thus we must have c < \sup \{ y \mid F(y) < \omega \}

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