Internal and spacetime symmetries? I am trying to understand Wiki's explanation about correlation. Part of this article talks about internal and spacetime symteries:

If the probability distribution has any target space symmetries, i.e.
  symmetries in the space of the stochastic variable (also called
  internal symmetries), then the correlation matrix will have induced
  symmetries. If there are symmetries of the space (or time) in which
  the random variables exist (also called spacetime symmetries) then the
  correlation matrix will have special properties.

And then it follows with two examples about symmetries. However I didn't quite understand the definition nor I know any intuitive example for each case?
Can somebody help me regarding this; first an explanation of definition and secondly some examples. What does it mean to say "target space" symmetries in space of stochastic variable and what is the difference of that to "symmetries of the space (or time) in which the random variables exist"? 
 A: Let $M$ be a manifold, let $V$ be a finite-dimensional vector space, and let $\Omega$ be a sample space (in the sense of probability).  For each point $p\in M$, let $X(p):\Omega\to V$ be a random variable.
A mapping $T_V:V\to V$ is called a target space transformation and a mapping $T_M:M\to M$ is called a space transformation (or spacetime transformation if $M$ can be thought of as a spacetime.)
Let $\rho:\Omega\to \mathbb R$ be a probability distribution on $\Omega$.  We call a function $T_\Omega:\Omega\to\Omega$ a sample space transformation.  We say that the measure $d\omega \rho(\omega)$ is invariant under this transformation provided $dT(\omega)\,\rho(T(\omega)) = d\omega\,\rho(\omega)$ for all $\omega\in \Omega$.  In this case $T_\omega$ is called a symmetry.  We define the correlation function $C^{ij}_X(p,q)$ as
\begin{align}
  C^{ij}_X(p,q) = \int_\Omega d\omega\,\rho(\omega) \,X^i(p)(\omega)X^j(q)(\omega)
\end{align}
where $X^i(p)$ is the $i^\mathrm{th}$ component of $X(p)$ when it is expanded in some basis of $V$.
Example. Conformal field theory on $\mathbb R^2$.
Let $M = \mathbb R^2$, $V = \mathbb R$, and $\Omega$ be a set of admissible functions $\phi:M\to V$.  This is the situation one would have in a quantum field theory of a real scalar field on the plane.  For each $\lambda\neq 0$ and each real $\Delta$, we define a scale transformation $T_\Lambda$ on $\Omega$ as
\begin{align}
  T_{\lambda,\Delta}(\phi)(\mathbf x)= \lambda^{-\Delta} \phi(\lambda^{-1}x)
\end{align}
which is simply the space transformation $\mathbf x\to \lambda^{-1}\mathbf x$ composed with the target space transformation $\phi(\mathbf x)\to \lambda^{-\Delta}\phi(\mathbf x)$.  Formally, there are certain distributions on $\Omega$ that exhibit so-called scale-invariance, namely invariance under $T_{\lambda, \Delta}$.  Let $X_\phi(\mathbf x)$ be the random variable
\begin{align}
  X(\mathbf x)(\phi) = \phi(\mathbf x)
\end{align}
then given scale-invariance, $T_\omega$ induces a scale transformation on the correlator which I leave it to you to verify.
\begin{align}
    C_X(\mathbf x, \mathbf y) \to \lambda^{-2\Delta}C_X(\lambda^{-1}\mathbf x, \lambda^{-1}\mathbf y).
\end{align}
