Derivation of n ≥ 2  p in the Champernowne Equation

Introduction

Champernowne (1948, eqn 3.5) showed that

(z^′) (Subscript[Γ, n]^(-1)) 
        z = (β^′) 
        D  β /(Subscript[σ, a]^2),

where \(TraditionalForm\`β = \((\(-1\), ϕ\_1, \  ... \ , \ ϕ\_p)\)\) and D is the (p + 1)-by-(p + 1) matrix with (i,  j)-entry

\(TraditionalForm\`D\_\(i, j\) = \(D\_\(j, \ i\) = \ \(\(\(z\_i\ z\_j\)\(\ \)\(+\)\)\  ... \)\  + \ 
          z\_\(n + 1 - i\)\ z\_\(n + 1 - j\)\)\)

Just below eqn (3.2), Champernowne (1948) indicates that n > 2  p.  Close examination of the derivation given in Champernowne (1948) and more elegantly by Box, Jenkins and Reinsel (1994, A7) indicates that a necessary and sufficient condition for (1) above to hold is that n ≥ 2  p.

Computation of D

GD[p_,  n_,  z_] := 
      Module[{}, 
𝔇 = 
          Array[0 &, {p + 1, 
              p + 1}]; 
Do[
𝔇[[i, j]] = 
            Sum[z[[i + k]] 
                z[[j + 
                    k]], 
{k, 0, 
                n + 1 - i - j}], {i, p + 1}, {j, 
            p + 1}]; 
𝔇]

z = Table[Subscript[𝓏, k], {k, 4}];

GD[2, 4, z]

<< FitAR.m

Exercise.  Take p=2 and n=5.

z = Range[5]

GD[2, 5, z]

β = {-1, 1.6,  -0.64};

β . 𝔇 . β

G = Toeplitz[TacvfAR[{1.6,  -0.64}, 4]];

z . Inverse[G] . z

Exercise.  Take p=2 and n=4.

z = Range[4]

GD[2, 4, z]

β = {-1, 1.6,  -0.64};

β . 𝔇 . β

G = Toeplitz[TacvfAR[{1.6,  -0.64}, 3]];

z . Inverse[G] . z

Exercise.  Take p=2 and n=3.

z = Range[3]

GD[2, 3, z]

β = {-1, 1.6,  -0.64};

β . 𝔇 . β

G = Toeplitz[TacvfAR[{1.6,  -0.64}, 2]];

z . Inverse[G] . z