Lemma
Proposition
Stein's Equation
All solutions of Stein's Equation are of the form
\[
f(x) = Ce^{x^2/2} + e^{x^2/2}\int_{-\infty}^x [h(y)-\mathbb{E}(h(N))]e^{y^2/2} dy
\]
In particular, denote by
\[
f_h(x) = e^{x^2/2}\int_{-\infty}^x [h(y)-\mathbb{E}(h(N))]e^{y^2/2} dy
\]
In particular, $f_h$ is he only solution such that
\[
\lim_{x \to \infty} e^{x^2/2} f_h(x) = 0
\]
Proof
$ \sqrt{2\pi}\int_{-\infty}^x h(y) \frac{e^{y^2/2}}{\sqrt{2\pi}} - \mathbb{E}(h(N))\int_{-/infty}^\infty \gamma(dx) = 0$
Depends on
Used in
Dependency Graph
flowchart LR
classDef current fill:#6366f1,color:#fff,stroke:#4f46e5
n19bfbd84["Stein's Equation"]
ncfde8c86["Proposition"]:::current
na8ae60fa["Stein Bounds for TV Metric"]
n19bfbd84 --> ncfde8c86
ncfde8c86 --> na8ae60fa
click n19bfbd84 "../objects/19bfbd84.html" "_self"
click na8ae60fa "../objects/a8ae60fa.html" "_self"