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Let $K(\cdot|\cdot)$ be a Markov kernel. For a measure probability $\mu$, denote by $\mu K$ the probability measure induced by $K$ and $\mu$, i.e. $(\mu K)(A):=\int_\Omega \mu(d\omega)K(A|\omega)$ for each (Borel) measurable $A$.

For an appropriate notion of "distance" $d$ between distributions, one would like a result saying that that whenever $d(\mu,\nu)\leq \epsilon$ then $d(\mu K,\nu K)\leq \epsilon\cdot C$, for some constant $C$ not depending on $\epsilon$.

The distance $d$ should satisfy some concentration bound implying, loosely speaking, that the distance between $\mu$ and the empirical measure obtained by $N$ i.i.d. samplings from $\mu$ approaches 0, as $N\rightarrow+\infty$.

I am aware of a result along these lines for the $\ell_1$-distance attributed to Dobrushin, see this question; but did not find a reference for that. Any (introductory) references to results concernig other types of distances would be much appreciated as well.

Let $K(\cdot\mid\cdot)$ be a Markov kernel. For a measure probability $\mu$, denote by $\mu K$ the probability measure induced by $K$ and $\mu$, i.e. $(\mu K)(A):=\int_\Omega \mu(d\omega)K(A\mid\omega)$ for each (Borel) measurable $A$.

For an appropriate notion of "distance" $d$ between distributions, one would like a result saying that that whenever $d(\mu,\nu)\leq \varepsilon$ then $d(\mu K,\nu K)\leq \varepsilon\cdot C$, for some constant $C$ not depending on $\varepsilon$.

The distance $d$ should satisfy some concentration bound implying, loosely speaking, that the distance between $\mu$ and the empirical measure obtained by $N$ i.i.d. samplings from $\mu$ approaches 0, as $N\rightarrow+\infty$.

I am aware of a result along these lines for the $\ell_1$-distance attributed to Dobrushin, see this question; but did not find a reference for that. Any (introductory) references to results concernig other types of distances would be much appreciated as well.

Let $K(\cdot|\cdot)$ be a Markov kernel. For a measure probability $\mu$, denote by $\mu K$ the probability measure induced by $K$ and $\mu$, i.e. $(\mu K)(A):=\int_\Omega \mu(d\omega)K(A|\omega)$ for each (Borel) measurable $A$.

For an appropriate notion of "distance" $d$ between distributions, one would like a result saying that that whenever $d(\mu,\nu)\leq \epsilon$ then $d(\mu K,\nu K)\leq \epsilon\cdot C$, for some constant $C$ not depending on $\epsilon$.

The distance $d$ should satisfy some concentration bound implying, looseley speaking, that the distance between $\mu$ and the empirical measure obtained by $N$ i.i.d. samplings of $\mu$ approaches 0, as $N\rightarrow+\infty$.

I am aware of a result along these lines for the $\ell_1$-distance attributed to Dobrushin, see this question; but did not find a reference for that. Any (introductory) references to results concernig other types of distances would be much appreciated as well.

Let $K(\cdot|\cdot)$ be a Markov kernel. For a measure probability $\mu$, denote by $\mu K$ the probability measure induced by $K$ and $\mu$, i.e. $(\mu K)(A):=\int_\Omega \mu(d\omega)K(A|\omega)$ for each (Borel) measurable $A$.

For an appropriate notion of "distance" $d$ between distributions, one would like a result saying that that whenever $d(\mu,\nu)\leq \epsilon$ then $d(\mu K,\nu K)\leq \epsilon\cdot C$, for some constant $C$ not depending on $\epsilon$.

The distance $d$ should satisfy some concentration bound implying, loosely speaking, that the distance between $\mu$ and the empirical measure obtained by $N$ i.i.d. samplings from $\mu$ approaches 0, as $N\rightarrow+\infty$.

I am aware of a result along these lines for the $\ell_1$-distance attributed to Dobrushin, see this question; but did not find a reference for that. Any (introductory) references to results concernig other types of distances would be much appreciated as well.

Let $K(\cdot|\cdot)$ be a Markov kernel. For a measure probability $\mu$, denote by $K\mu$ the probability measure induced by $K$ and $\mu$, i.e. $(K\mu)(A):=\int_\Omega \mu(d\omega)K(A|\omega)$ for each (Borel) measurable $A$.

For an appropriate notion of "distance" $d$ between distributions, one would like a result saying that that whenever $d(\mu,\nu)\leq \epsilon$ then $d(K\mu,K\nu)\leq \epsilon\cdot C$, for some constant $C$ not depending on $\epsilon$.

Also, convergence in distance $d$ should imply convergence of measures in an appropriate metric.

I am aware of a result along these lines for the $\ell_1$-distance attributed to Dobrushin, see this question; but did not find a reference for that. Any (introductory) references to results concernig other types of distances would be much appreciated as well.

Let $K(\cdot|\cdot)$ be a Markov kernel. For a measure probability $\mu$, denote by $\mu K$ the probability measure induced by $K$ and $\mu$, i.e. $(\mu K)(A):=\int_\Omega \mu(d\omega)K(A|\omega)$ for each (Borel) measurable $A$.

For an appropriate notion of "distance" $d$ between distributions, one would like a result saying that that whenever $d(\mu,\nu)\leq \epsilon$ then $d(\mu K,\nu K)\leq \epsilon\cdot C$, for some constant $C$ not depending on $\epsilon$.

The distance $d$ should satisfy some concentration bound implying, looseley speaking, that the distance between $\mu$ and the empirical measure obtained by $N$ i.i.d. samplings of $\mu$ approaches 0, as $N\rightarrow+\infty$.

I am aware of a result along these lines for the $\ell_1$-distance attributed to Dobrushin, see this question; but did not find a reference for that. Any (introductory) references to results concernig other types of distances would be much appreciated as well.