61. Finite moments cannot validate an unrestricted density class
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A finite observation basis is a modeling assumption. Positivity certification does not turn that assumption into a consequence of data.
Proposition 55: A valid continuous counterexample to finite-moment identification
Let
Relative to \(du/2\), these are distinct strictly positive normalized densities with the same moments of orders zero, one and two. Their total-variation distance is
Taking independent observations gives two rank-one processes whose products of these low-order moments agree at every sequence length.
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Proof
The endpoints and stationary points of \(L_3\) show \(|L_3(u)|\leq1\) on the interval. Direct integration gives \(\int u^jL_3(u)\,du/2=0\) for \(j=0,1,2\). This proves normalization and moment agreement, while strict positivity follows from \(\epsilon<1\). Its nonnegative root other than zero is \(\sqrt{3/5}\). Integrating \(|L_3|\) on the two subintervals of \([0,1]\) and using symmetry gives \(\int|L_3|\,du/2=13/40\). The density difference is \(2\epsilon L_3\), so its half-integral is the stated distance. Independence makes every product moment factor into the identical one-step moments. \(\square\)
This is not a defect in Proposition 54. These densities lie outside its quadratic class. It is a concrete reason to retain an explicit approximation or misspecification term when increasing the observation basis. Internal probability validity and correct density identification remain different claims even in one predictive dimension.