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Baby Skorohod Theorem

Last updated Nov 1, 2022

# Statement

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

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

# Proof

# Other Outlinks

Convergence in Distribution
Almost Sure Convergence

Almost Sure Convergence

Definition Suppose $(\Omega, \mathcal{M}, \mathbb{P})$ is a and $(Y, \tau)$ is a . Suppose $(X{n} : \Omega \to Y)$ is...

11/7/2022

Backlinks

Continuous Mapping Theorem

Continuous Mapping Theorem

...Let $h: \mathbb{R} \to \mathbb{R}$ be s.t. $\mathbb{P}[X_{0} \in \text{Disc}(h)] = 0.$ Then $h(X{n}) \Rightarrow h(X{0}).$ If $h$ is bounde, then $\mathbb{E}(h(X{n})) = \int\limits h(x) F{n}(dx) \to \mathbb{E}(h(X{0})) = \int\limits h(x) F{0}(dx)$ Proof See Cor 8.3.1 pg......

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