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.

  • $\begingroup$ Do you mean GPC (generalized polynomial chaos) expansions up to English grammar? $\endgroup$ – user64494 Aug 3 '19 at 11:13
  • $\begingroup$ @user64494 It is always written as gPC (look up any reference on uncertainty quantification, for example the book I mentioned in the question). I do not know the exact reason. There are PC (polynomial chaos) expansions, which are the polynomial expansions arising from $Z$ Gaussian and $\{\phi_m(Z)\}_{m=0}^\infty$ Hermite polynomials. Then Xiu and Karniadakis generalized PC expansions to distributions of $Z$ that belong to the Wiener-Askey scheme. This gave rise to gPC (generalized polynomial chaos) expansions. $\endgroup$ – jxm Aug 3 '19 at 13:39

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