WeiYa's Work Yard

A dog, who fell into the ocean of statistics, tries to write down his ideas and notes to save himself.

Noise Outsourcing

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Tags: Probability

I learnt the term Noise Outsourcing in kjytay’s blog, which is based on Teh Yee Whye’s IMS Medallion Lecture at JSM 2019.

(Noise Outsourcing). If $X$ and $Y$ are random variables in "nice" (e.g. "Borel") spaces $\cX$ and $\cY$, then there is a random variable $\eta\sim U[0,1]$ which is independent of $X$ and a function $h:[0,1]\times \cX\mapsto \cY$ such that $$ (X,Y)\overset{a.s.}{=}(X, h(\eta, X))\,. $$ In particular, if there is a statistic $S(X)$ with $X\ind Y\mid S(X)$, then $$ (X,Y)\overset{a.s.}{=}(X, h(\eta, S(X)))\,. $$

In their arxiv’s words, Noise Outsourcing is a “standard technical tool from measure theoretic probability”, where it is also known by other names such as transfer. Kallenberg (2002) adopted the term transfer,

(transfer). Fix any measurable space $S$ and Borel space $T$, and let $\xi\overset{d}{=}\tilde \xi$ and $\eta$ be random elements in $S$ and $T$, respectively. Then there exists a random element $\tilde \eta$ in $T$ with $(\tilde \xi, \tilde \eta)\overset{d}{=}(\xi,\eta)$. More precisely, there exists a measurable function $f:S\times [0,1]\rightarrow T$ such that we may take $\tilde \eta=f(\tilde \xi, \vartheta)$ whenever $\vartheta\ind \tilde\xi$ is $U(0,1)$.

and said that, the basic transfer theorem may be used to convert any distributional equivalence $\xi\overset{d}{=}f(\eta)$ into a corresponding a.s. representation $\xi=f(\tilde \eta)$ with a suitable $\tilde \eta\overset{d}{=}\eta$.

It reminds me a theorem learnt in the advanced probability course with Durrett Rick’s Probability: Theory and Examples (4th ed),

If $F_n\rightarrow_d F_\infty$, then there are random variables $Y_n, 1\le n\le \infty$ with distribution $F_n$ so that $F_n\rightarrow_{a.s.}Y_\infty$.

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