Suppose that a random variable $Y$ can be written as $Y=g(Z)$, where $g$ is a function and $Z$ is a random variable. When $Z$ is a continuous random variable with finite absolute moments, we consider a sequence of orthogonal polynomials with respect to the density function $f_Z$, $\{\phi_m(Z)\}_{m=0}^\infty$, which is called the generalized polynomial chaos (gPC) basis. Then $Y$ has the following gPC expansion:
$$ Y= \sum_{m=0}^\infty y_m \phi_m(Z), \quad y_m=\frac{E[f(Z)\phi_m(Z)]}{E[\phi_m(Z)^2]}. $$
These expansions can be generalized to random vectors $Z$ and have lots of applications when solving stochastic systems. An introduction is presented in the book *Numerical Methods for Stochastic Computations, A Spectral Method Approach*, by Dongbin Xiu (2010).

As discussed in the book, convergence of gPC expansions holds in the mean square sense when the support of $Z$ is bounded. Moreover, this convergence is spectral (the smoother $g$ is, faster convergence holds; if $g$ is analytic, exponential convergence holds). In the paper *On the convergence of generalized polynomial chaos expansions*, ESAIM: M2AN 46 (2012) 317-339, it is proved that mean square convergence holds when the moment problem for $Z$ is uniquely solvable.

My question is whether there is any theoretical result in the literature that guarantees convergence of gPC expansions in the total variation distance.