On Realizable Distributions

Siddharth Bhandari
Saturday, 11 Jul 2020, 16:00 to 17:00
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Meeting ID: 981 7222 2418
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Abstract: Coupling arguments are commonplace in approximate and perfect sampling. A coupling is a joint distribution between RVs: it helps us relate and compare distributions of the RVs.

We will study the following:
Given RVs X_1, X_2, X_3, X_4,...., X_n each taking a value in $\Omega$ and marginally distributed according to $\mu_1, \mu_2, ..., \mu_n$ on $\Omega^n$, we say a RV $Y$ distributed according to $\tau$ is realizable wrt to $X_1,..., X_n$ iff for all joint distributions $\mu$ of $X_1,.., X_n$ such that each $X_i~\mu_i$ there exists a joint distribution  $\mu'$ of $X_1,..., X_n, Y$ such that 1> $(X_1, X_2, ..., X_n)~\mu$  and $Y~\tau$ 2> Pr_{\mu'}[Y\in {X_1,..., X_n}]=1.
We will show that  a distribution $\tau$ is realizable wrt to $\mu_1,.., \mu_n$ iff for all $S\subseteq \Omega$ we have: $\mu_i(S)\leq \tau(S)\leq \mu_j (S)$ for some $i,j\in [n]$.