Let $g:R^{n+m} \to R$ be a deterministic function of some independent random variables $x_1,\ldots,x_n$ with distributions $f_{x_1}(x),\ldots,f_{x_n}(x)$ and some deterministic variables $z_1,\ldots,z_m$. Let $y_i$ be a random variable with the same support as $x_i$. Let $D(f_{x_i},f_{y_i})$ be a distance measure of two distributions. Given $D(f_{x_i},f_{y_i}) < \epsilon$, can we bound the following quantity \begin{equation} D(f_{g(x_1,\ldots,x_i,\ldots,x_n,z_1,\ldots,z_m)},f_{g(x_1,\ldots,y_i,\ldots,x_n,z_1,\ldots,z_m)}) \end{equation} Feel free to consider any reasonable assumptions on $f_{x_i}$, $g$, and perhaps $D$.
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$\begingroup$ $L^1$ distance (equivalently, twice the total variation or statistical distance) is good for this  if $\f_{x_i}  f_{y_i}\_1 \leq \epsilon$, then the quantity you want is bounded by $\epsilon$  should be easy to show using definition of total variation distance. $\endgroup$ – usul Oct 17 '16 at 23:08