Suppose a random variable $ Y $ can be written as $ Y = g (Z) $, from where $ g $ is a function and $ Z $ is a random variable. When $ Z $ If 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 $This is called a generalized polynomial chaos (gPC) basis. Then $ Y $ has the following gPC extension:

$$ Y = sum_ {m = 0} ^ infty y_m phi_m (Z), quad y_m = frac {E (f (Z) phi_m (Z))} {E ( phi_m (Z) ^ 2)}. $$

These extensions can be generalized to random vectors $ Z $ and have many applications in solving stochastic systems. An introduction will be presented in the book *Numerical methods for stochastic calculations, a spectral method approach*, by Dongbin Xiu (2010).

As discussed in the book, the convergence of gPC extensions in the middle quadratic sense applies when supporting $ Z $ is limited. In addition, this convergence is spectral (the smoother $ g $ is, holds a faster convergence; if $ g $ is analytic, exponential convergence applies). In the newspaper *About the convergence of generalized polynomial chaos extensions*, ESAIM: M2AN 46 (2012) 317-339, it is proved that the mean square convergence holds when the instantaneous problem for $ Z $ is clearly solvable.

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