Baby Skorohod Theorem

Last updated Nov 1, 2022

Let ${X_{n} : n \geq 0}$ be a Sequence of Random Variables (not necessarily defined on the same Probability Space) so that $X_{n} \Rightarrow X_{0}$. Then there exist ${X^{#}_{n} : n \geq 0 }$ defined on Probability Space $([0,1], \mathcal{B}([0,1]), \lambda)$ (where $\lambda$ is the Lebesgue Measure) so that

$$\begin{align*} &X_{n}^{#} \overset{d}= X_{n} \text{ for } n \geq 0\\ &X_{n}^{#} \to X_{0}^{#} \text{ a.s.}\\ \end{align*}$$

• Convergence in Distribution
• Almost Sure Convergence