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Baby Skorohod Theorem

Last updated Nov 1, 2022

# Statement

Let Xn:n0{X_{n} : n \geq 0} be a Sequence

Sequence

Definition A f:NXf: \mathbb{N} \to X for some XX. It is usually denoted {xn}n=1X\{xn\}{n=1}^{\infty} \subset X or $(x{n}) \subset...

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of Random Variable

Random Variable

Definition Let (Ω,B,P)(\Omega, \mathcal{B}, \mathbb{P}) be a . A is a X:ΩRX: \Omega \to \mathbb{R}. Remarks Rather than say XX is...

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s (not necessarily defined on the same Probability Space

Probability Space

...

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) so that XnX0X_{n} \Rightarrow X_{0}. Then there exist {X^{#}_{n} : n \geq 0 } defined on Probability Space

Probability Space

...

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([0,1],B([0,1]),λ)([0,1], \mathcal{B}([0,1]), \lambda) (where λ\lambda is the Lebesgue Measure

Lebesgue Measure

Definition The with idR\text{id}_\mathbb{R}. Other Outlinks ...

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) 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