I have some questions about Chapter 5 from the book Numerical Methods for Stochastic Computations, by Dongbin Xiu.
Theorem 5.7: Let $Y$ be a random variable and $\mathbb{E}[Y^2]<\infty$. Let $Z$ be random variable and $\mathbb{E}[Z^{2m}]<\infty$ for all $m\geq0$ with generalized polynomial chaos basis $\{\Phi_k(Z)\}_{k=0}^\infty$, which satisfies $\mathbb{E}[\Phi_m(Z)\Phi_n(Z)]=\delta_{mn}\gamma_n$, $m,n\geq0$. Let $Y_N=\sum_{k=0}^N a_K \Phi_k(Z)$, where $a_k=\frac{1}{\gamma_k}\mathbb{E}[F_Y^{-1}(F_Z(Z))\Phi_k(Z)]$, $0\leq k\leq N$. Then $Y_N$ converges to $Y$ in probability.
Proof: Let $\tilde{Y}:=G(Z)=F_Y^{-1}(F_Z(Z))$. Then $\tilde{Y}$ and $Y$ have the same distribution, so $\mathbb{E}[\tilde{Y}^2]<\infty$. Since $Y_N$ is the orthogonal projection of $\tilde{Y}$, then $Y_N\rightarrow \tilde{Y}$ in $L^2$. This implies that $Y_N\rightarrow \tilde{Y}$ in probability. Since $Y$ and $\tilde{Y}$ have the same distribution, we conclude that $Y_N\rightarrow Y$ in probability.
Question 1: Why do we need $\mathbb{E}[Z^{2m}]<\infty$ ? On the other hand, I do not see why $Y_N\rightarrow Y$ in probability in the last reasoning.
In page 64, the author defines generalized polynomial chaos basis for random vectors $Z=(Z_1,\ldots,Z_d)$ with independent components. Let $\{\phi_k(Z_i)\}_{k=0}^\infty$ be a generalized polynomial chaos basis for $Z_i$: $\mathbb{E}[\phi_m(Z_i)\phi_n(Z_i)]=\delta_{mn}\gamma_m$. Let $\mathbb{i}=(i_1,\ldots,i_d)$ be a multi-index. Let $\Phi_{\mathbb{i}}(Z)=\phi_{i_1}(Z_1)\cdots\phi_{i_d}(Z_d)$, $0\leq|\mathbb{i}|\leq N$. Then $\{\Phi_{\mathbb{i}}(Z)\}_{\mathbb{i}}$ is the generalized polynomial chaos basis for $Z$.
Question 2: I think that each sequence $\{\phi_k(Z_i)\}_{k=0}^\infty$ should depend on $i$, as the orthogonality depends on the law of $Z_i$: $\{\phi_k^i(Z_i)\}_{k=0}^\infty$, $\Phi_{\mathbb{i}}(Z)=\phi_{i_1}^1(Z_1)\cdots \phi_{i_d}^d(Z_d)$. Is this correct?
Question 3: In Table 5.2, there is an example of graded lexicographic order for multi-indexes of dimension 4: $(0,0,0,0)$ is in position 1, $(1,0,0,0)$ is in position 2, $(0,1,0,0)$ is in position 3, etc. According to the definition of this order, $\mathbb{i}>\mathbb{j}$ if and only if $|\mathbb{i}|\geq|\mathbb{j}|$ and the first nonzero entry of $\mathbb{i}-\mathbb{j}$ is positive. For example, we have $(0,0,0,0)<(1,0,0,0)$. However, $(0,1,0,0)<(1,0,0,0)$, but in the table this order is reversed. Is this correct?
Question 4: In page 67, it is said the generalized polynomial chaos allows approximating the expectation and covariance of $f(t,Z)$, where $Z$ is a random vector and $t\in T$, $T$ being an index set. It is written $f_N(t,Z)=\sum_{|\mathbb{i}|\leq N}\hat{f}_{\mathbb{i}}(t)\Phi_{\mathbb{i}}(Z)$ as the approximation of $f(t,Z)$ in $L^2$. Then, in formula (5.33), it is written $\mathbb{E}[f(t,Z)]\approx \mathbb{E}[f_N(t,Z)]=\hat{f}_{\mathbb{0}}(t)$. But $\hat{f}_{\mathbb{0}}(t)$ is precisely $\mathbb{E}[f(t,Z)]$, so the approximation should be in fact an equality. Moreover, to compute the approximation $f_N(t,Z)$, we need the Fourier coefficients, so we need $\hat{f}_{\mathbb{0}}(t)=\mathbb{E}[f(t,Z)]$, which is what we wanted to approximate.
