# Spectral expansion of a discrete function of a set of continuous random variables

Let $\mathbf{X}(\omega)=(X_1(\omega), \dots, X_n(\omega))^T$, $\omega \in \Omega$, be a vector of independent continuous random variables defined on a probability space $(\Omega, \mathcal{F}, \mathcal{P})$. Also, let $f(\omega) = \sum_{i = 1}^m \alpha_i \mathbf{1}_{A_i}(\omega)$ be a simple function where $\mathbf{1}_{A_i}(\omega)$ are indicator functions, $A_i = \{ \omega: g_i(\mathbf{X}(\omega)) = 0 \} \in \mathcal{F}$ are disjoint events, and $g_i(\mathbf{X}(\omega))$ are given functions of the random variables. According to the definition, $f(\omega)$ is a discrete random variable, which takes the values $\alpha_i$ with certain probabilities $p_i$, $i \in \{1, \dots, m\}$.

The question is in the following: Is it possible to construct an orthogonal expansion of $f(\omega)$ in terms of $\mathbf{X}(\omega)$? I am familiar with the theory of generalized polynomial chaos (gPC) expansions. It works pretty well with nice functions, which can be approximated by polynomials. But in this case we have an arbitrary discrete probability distribution, and $X_i(\omega)$ are continuous. I am not sure that a non-intrusive spectral projection would give any meaningful results. I am aware that the gPC handles certain types of discrete probability distributions, but here I am a bit lost: what could be a suitable basis in this case?

Thank you.

Regards, Ivan

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