Skorohod Representation Theorem

1 Skorohod 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 \(n \to \infty\).

1.0.1 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 \(Y(\omega):= \inf\{ x: \omega \leq F(x) \}\).

Skorohod Construction

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 \[ \begin{align} \mathbb{P}'(Y_{n}(\omega)\leq y) =\mathbb{P}'(\omega \leq F_{n}(y)) & =\mathbb{P}'([0,F_{n}(y)])=F_{n}(y) \\ \mathbb{P}'(Y(\omega)\leq y)& = F(y). \end{align} \] Proof of Skorohod Representation Theorem: Given \(X_{n}\Rightarrow X\) we show that \(Y_{n}\to Y\) a.s.. Recall that