Each \(E_j\) is symmetric, because its matrix is constant on the
pairs inside a class and zero across classes, and \(E_j^2=E_j\),
because a class mean is unchanged by a second average. If \(i\ge j\),
congruence modulo \(p^{k-i}\) is coarser than congruence modulo
\(p^{k-j}\), so each coarser class is a union of finer classes. A
function constant on a coarser class is constant on each finer class
inside it, and therefore \(E_jE_i=E_i\). That identity does not use the
sizes of the classes. For the other order, every finer class has
\(p^j\) sites, so the mean on a coarser class is the mean of the finer
means and \(E_iE_j=E_i\).
In particular \(E_{j+1}E_j=E_jE_{j+1}=E_{j+1}\). Then
\(P_jE_{j+1}=E_jE_{j+1}-E_{j+1}^2=E_{j+1}-E_{j+1}=0\). If \(i>j\),
then \(i\ge j+1\).
The size-free identity gives \(E_{j+1}E_i=E_i\), so
\(E_if=E_{j+1}(E_if)\) and the image of \(E_i\) lies in the image of
\(E_{j+1}\). When \(i<k\) the same left factor gives
\(E_{j+1}E_{i+1}=E_{i+1}\), and both summands of \(P_i\) land in that
image. When \(i=k\), \(E_{j+1}E_k=E_k\). A vector in the image of
\(P_i\) is therefore \(E_{j+1}\) applied to something, and
\(P_jE_{j+1}=0\) gives \(P_jP_i=0\). Both matrices are symmetric, so the
opposite product vanishes as well. For \(j<k\) the square expands to
\(E_j-E_jE_{j+1}-E_{j+1}E_j+E_{j+1}^2\), which equals
\(E_j-E_{j+1}-E_{j+1}+E_{j+1}=P_j\). The case \(P_k=E_k\) is symmetric
and idempotent because \(E_k\) is. The sum of the projectors telescopes
to \(E_0\). Thus the \(P_j\) are pairwise orthogonal projectors summing
to the identity, and \(Q\) multiplies the image of \(P_j\) by \(c_j\).
Its inverse multiplies that image by \(c_j^{-1}\). The functions carry
the counting inner product. Since \(P_j\) is
symmetric and idempotent, \(\xi \cdot P_j\xi =|P_j\xi |^2\), and the quadratic
form is \(\xi \cdot Q\xi =\sum _j c_j|P_j\xi |^2\). If \(\xi \neq 0\), some
\(P_j\xi \) is nonzero because the projectors sum to the identity. Whenever
every coefficient is positive, the form is therefore positive, and it
is the precision of a centered Gaussian. By the
inverse-precision theorem, that law has covariance
operator \(Q^{-1}\). In the counting basis, the second-moment matrix of
\(A\xi \) is \(AQ^{-1}A^{T}\).
Write \(S\) for the lift that makes a coarse function constant on
each class:
\((Sg)(r+tp^{k-1})=g(r)\). Averaging a constant returns the constant,
so \(AS=I\). The counting pairing gives
\((A^{T}g)\cdot \xi =g\cdot A\xi \), and the right-hand side expands to
\[\sum _r g(r)\frac {1}{p}\sum _{t=0}^{p-1}\xi (r+tp^{k-1}).\]
The coefficient of \(\xi (r+tp^{k-1})\) is \(g(r)/p\), so
\(A^{T}=S/p\). The class mean is
\((E_1\xi )(r+tp^{k-1})=(A\xi )(r)\). Averaging those \(p\) equal values
returns \((A\xi )(r)\), hence \(AE_1=A\) and \(AP_0=0\).
For \(j\ge 1\), the coarse average \(E'_{j-1}\) is formed on the
residues modulo \(p^{k-1}\) by classes congruent modulo \(p^{k-j}\).
Fix a site \(x=r+tp^{k-1}\). Since \(j\ge 1\), the step \(p^{k-1}\) is
divisible by \(p^{k-j}\), so the fine class of \(x\) modulo \(p^{k-j}\)
consists of the sites \(r'+t'p^{k-1}\) whose coarse residue satisfies
\(r'\equiv r\) modulo \(p^{k-j}\). Each such \(r'\) occurs for all \(p\)
choices of \(t'\), and \(Sg\) depends only on \(r'\). The fine mean is
therefore the coarse mean of \(g\), which is
\((E'_{j-1}g)(r)=(SE'_{j-1}g)(x)\). Thus \(E_jS=SE'_{j-1}\). Applying
\(A\) and using \(AS=I\) gives \(AE_jS=E'_{j-1}\), and
\(AE_jA^{T}=E'_{j-1}/p\).
For \(1\le j\le k-1\) this difference is
\(AP_jA^{T}=(E'_{j-1}-E'_j)/p\). The top term is
\(AE_kA^{T}=E'_{k-1}/p\). Because \(AP_0=0\),
\[AQ^{-1}A^{T}=\sum _{j=1}^{k}c_j^{-1}AP_jA^{T}.\]
Reindex \(i=j-1\). The right-hand side equals
\[\sum _{i=0}^{k-1}(pc_{i+1})^{-1}P'_i,\]
where \(P'_i=E'_i-E'_{i+1}\) for \(i<k-1\) and \(P'_{k-1}=E'_{k-1}\).
The same orthogonal decomposition on the coarser residues says that
this sum is the inverse of the hierarchical precision \(Q'\) with
coefficients \(c'_i=pc_{i+1}\). The matrix \(AQ^{-1}A^{T}\) is therefore
invertible, and its inverse is \(Q'\). If the fine coefficients are
positive, the coarse coefficients are positive as well.
Now take \(c_j=\lambda ^j\) for \(j<k\) and \(c_k=m\), with \(\lambda >0\) and
\(m>0\). Then \(c'_i=p\lambda ^{i+1}\) for \(i<k-1\), while
\(c'_{k-1}=pm\). Dividing \(Q'\) by \(p\lambda \) sends \(c'_i\) to \(\lambda ^i\) below
the top level and sends the top coefficient to \(m/\lambda \). If \(Z\) denotes
\(\sqrt {p\lambda }\,A\xi \), the second-moment matrix of \(Z\) is \(p\lambda \) times the
second-moment matrix of \(A\xi \), so the second-moment matrix of
\(Z\) is the inverse of \(Q'/(p\lambda )\).
The top coefficient \(m/\lambda \) is larger than \(m\) when \(0<\lambda <1\), equal
to \(m\) when \(\lambda =1\), and smaller than \(m\) when \(\lambda >1\).
Repeat the construction at every depth. If positive coefficients
\(c^{(k)}_0,\ldots ,c^{(k)}_k\) satisfy
\(c^{(k-1)}_i=pc^{(k)}_{i+1}\) for each \(i\), the second-moment matrix
of the depth-\(k\) block mean is the inverse of the depth-\((k-1)\)
hierarchical precision. The same matrix identity repeats at every
depth. These identities do not by themselves produce a Gaussian
measure on functions on the \(p\)-adic integers.