0.1 Skorohod’s Representation Theorem
Skorohod’s Representation Theorem is a result showing that a convergent in law sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution / law of a pointwise convergent sequence of random variables defined on a common probability space.
Suppose that \(X_{n}\stackrel{\mathcal{D}}{\to}X\) as \(n \to \infty\) with \(F_{n}(x):=\mathbb{P}(X_{n} \leq x)\) and \(F(x):=\mathbb{P}(X\leq x)\) for \(x \in \mathbb{R}\). Then, there exists a probability space \((\Omega', \mathcal{F}',\mathbb{P}')\) and random variables \(\{Y_{n}, n\geq 1\}\) and \(Y\) such that \[ Y_{n}\stackrel{\mathcal{D}}{=}X_{n},\quad Y\stackrel{\mathcal{D}}{=X}, \] for every \(n\) and \(Y_{n}\stackrel{a.s.}{\to}Y\) as \(n \to \infty\).
0.2 Skorohod’s Construction
We take \(\Omega'=[0,1]\), \(\mathcal{F}'=\mathcal{B}([0,1])\) and \(\mathbb{P}'=\) the Lebesgue measure on \([0,1]\) \[ \mathbb{P}'((a,b])=b-a,~\forall a<b\in[0,1]. \] For each \(\omega \in \Omega'\) we define \(Y_n(\omega):= \inf\{ x: \omega \leq F_{n}(x)\}\) and \(Y(\omega):= \inf\{ x: \omega \leq F(x) \}\).
Note that \[ \begin{align} \omega \leq F_{n}(x) &~ \iff Y_{n}(\omega)\leq x \\ \omega \leq F(x) & ~ \iff Y(\omega)\leq x, \end{align} \] hence we have \[ \begin{align} \mathbb{P}'(Y_{n}(\omega)\leq y) =\mathbb{P}'(\omega \leq F_{n}(y)) & =\mathbb{P}'([0,F_{n}(y)])=F_{n}(y) \\ \mathbb{P}'(Y(\omega)\leq y)& = F(y). \end{align} \]
For a bounded continuous function \(\varphi(\cdot)\), if \(X_{n}\), \(n \geq 1\) conveges to \(X\) in distribution, as \(n\to \infty\), then \[ \lim_{n\to \infty}\mathbb{E}[\varphi(X_{n})]=\mathbb{E}[\varphi(X)]. \]