The triple $(\Omega, \mathcal{F}, P)$ is a probability space if
• $\Omega$ is the sample space, that is some possibly abstract set.
• $\mathcal{F}$ is a $\sigma$-algebra of sets - the measurable subsets o…
A random variable $X$ is a measurable function from the sample space $\Omega$ to $\R$
$$ X : \Omega \to \R $$
that is, the inverse of any Borel Set in $\R$ is $\mathcal{F}$-measurable:
$$ X^{-1} (A) = \{\omega : X(…
A real valued random variable $N$ has the standard Gaussian distribution if and only if for every test function
$f : \mathbb{R} \to \mathbb{R}$ that is differentiable with $f' \in \mathcal{L}'(\gamma)$, the expe…
Let $F$ be a random variable such that
\[
\mathbb{E}(f'(F)-Ff(f)) \approx 0
\]
for a large class of test functions $f$ we want to say that this is close to the standard Gaussian.
\[
L(F) \approx N(0,…
Let $N \sim N(0,1)$. Let $h : \mathbb{R} \to \mathbb{R}$ be a Borel function such that
\[
\mathbb{E}(|h(N)|) < \infty
\]
or in other words
\[
h \in \mathcal{L}(\gamma)
\]
The Stein Equation a…
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_{-\i…
Let $\mathcal{H}$ be a seperating class and let $h \in \mathcal{H}$.
Let $f_h$ be the solution to the Stein Equation associated with $h$. Then
\[
f'_h(x)-xf_h(x) = h(x)-\mathbb{E}(h(N))
\]
Let $…
The total variation metric is is defined as follows:
\[
d_\operatorname{TV}(F,G) = \sup_{B \in \mathcal{B}(\mathbb{R})} | \mathbb{P}(F \in B) - \mathbb{P}(G \in B)
\]
where $\mathcal{B}(\mathbb{R})$ are the Borel …
The Kolmogorow Metric metrizes the space of probability distributions given the following definition:
\[
d_{\operatorname{Kol}}(F,G) = \sup | P(F \le z) - P(G\le z) |
\]
We take the seperating class that defined the total variation metric :
\[
\mathcal{H}_{TV} = \{\mathbb{1}_B : B \in \mathcal{B}(\mathbb{R})\}
\]
The proposition is as follows:
Let $h…