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S.Surace
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Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$. Define $\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\int_X\kappa(dy|x)\nu(dx)$.

Denote by $\Gamma(\mu,\nu)$ the set of couplings of $\mu$ and $\nu$ and by $\Gamma(\tilde\mu,\tilde\nu)$ the corresponding set for $\tilde\mu$ and $\tilde\nu$.

Furthermore, let $S_{\kappa}$$S_{\kappa}:\Gamma(\mu,\nu)\to\Gamma(\tilde\mu,\tilde\nu)$ be the natural action of $\kappa$ on $\Gamma(\mu,\nu)$:mapping defined by

$$S_{\kappa}(\pi)(dy_1,dy_2)=\int_{X\times X}\kappa(dy_1|x_1)\kappa(dy_2|x_2)\pi(dx_1,dx_2),\quad \pi\in\Gamma(\mu,\nu).$$

Question: Is $S_{\kappa}$ onto $\Gamma(\tilde\mu,\tilde\nu)$?

Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$. Define $\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\int_X\kappa(dy|x)\nu(dx)$.

Denote by $\Gamma(\mu,\nu)$ the set of couplings of $\mu$ and $\nu$ and by $\Gamma(\tilde\mu,\tilde\nu)$ the corresponding set for $\tilde\mu$ and $\tilde\nu$.

Furthermore, let $S_{\kappa}$ be the natural action of $\kappa$ on $\Gamma(\mu,\nu)$:

$$S_{\kappa}(\pi)(dy_1,dy_2)=\int_{X\times X}\kappa(dy_1|x_1)\kappa(dy_2|x_2)\pi(dx_1,dx_2),\quad \pi\in\Gamma(\mu,\nu).$$

Question: Is $S_{\kappa}$ onto $\Gamma(\tilde\mu,\tilde\nu)$?

Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$. Define $\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\int_X\kappa(dy|x)\nu(dx)$.

Denote by $\Gamma(\mu,\nu)$ the set of couplings of $\mu$ and $\nu$ and by $\Gamma(\tilde\mu,\tilde\nu)$ the corresponding set for $\tilde\mu$ and $\tilde\nu$.

Furthermore, let $S_{\kappa}:\Gamma(\mu,\nu)\to\Gamma(\tilde\mu,\tilde\nu)$ be the mapping defined by

$$S_{\kappa}(\pi)(dy_1,dy_2)=\int_{X\times X}\kappa(dy_1|x_1)\kappa(dy_2|x_2)\pi(dx_1,dx_2),\quad \pi\in\Gamma(\mu,\nu).$$

Question: Is $S_{\kappa}$ onto $\Gamma(\tilde\mu,\tilde\nu)$?

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S.Surace
  • 1.7k
  • 11
  • 22

Is there a coupling that induces a given coupling via a transition kernel?

Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$. Define $\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\int_X\kappa(dy|x)\nu(dx)$.

Denote by $\Gamma(\mu,\nu)$ the set of couplings of $\mu$ and $\nu$ and by $\Gamma(\tilde\mu,\tilde\nu)$ the corresponding set for $\tilde\mu$ and $\tilde\nu$.

Furthermore, let $S_{\kappa}$ be the natural action of $\kappa$ on $\Gamma(\mu,\nu)$:

$$S_{\kappa}(\pi)(dy_1,dy_2)=\int_{X\times X}\kappa(dy_1|x_1)\kappa(dy_2|x_2)\pi(dx_1,dx_2),\quad \pi\in\Gamma(\mu,\nu).$$

Question: Is $S_{\kappa}$ onto $\Gamma(\tilde\mu,\tilde\nu)$?