Mixed fourth moment of thermal inputs

The mixed fourth moment of thermal inputs measures how squared temperature and expansion-coefficient deviations occur together. For centered inputs $A=a-\mathbb{E}a$ and $T=\theta -\mathbb{E}\theta$, defineIt has total input degree four. It is not the fourth moment of the product $AT$, which would involve $\mathbb{E}[A^4{T}^4]$. For a centered thermal product, the exact variance isConsequently individual input variances and their covariance do not in general determine the product variance.

For jointly normal centered inputs, write $A=s_{a}U$ and $T=s_T(\rho U+\sqrt{1-{\rho }^2}V)$ using independent standard normal distributions. Expanding and using $\mathbb{E}{U}^4=3$, $\mathbb{E}{U}^2{V}^2=1$ and vanishing odd terms givesThis is a specific joint-Gaussian calculation, consistent with the Isserlis theorem; normal-looking marginals alone do not justify it.

An original bounded counterexample illustrates the extra information. Let $X$ equal $-\sqrt{2},0,+\sqrt{2}$ with probabilities $1/4,1/2,1/4$, and let an independent sign $B$ take $\pm 1$ equally. Put $A=s_{a}X$ and $T=s_{T}BX$. Both standardized marginals have mean zero and variance one, and $\mathbb{E}(AT)=0$, but $M_{22}=2s_a^2{s}_T^2$. Replacing the second standardized input by an independent copy of $X$ preserves both marginals and zero covariance while changing $M_{22}$ to $s_a^2{s}_T^2$.

For an illustrative length correction $d=LAT$, take $L=100\mathrm{mm}$, $s_a=10^{-6}{\mathrm{K}}^{-1}$ and $s_T=0.1\mathrm{K}$. The two bounded models give standard deviations approximately14.14nm and10nm. Their factor-of-two variance difference comes entirely from joint occurrence, rather than changed marginal uncertainties.

NIST's thermal correction combines block length, expansion coefficient and temperature difference, providing the physical setting for this joint-input issue.^[ref] Joint observations and the way coefficient or thermometer corrections are assigned must support the dependence model. Increasing the number of simulated draws cannot recover an unmeasured mixed moment. The GUM explicitly calls for higher-order treatment when a nonlinear response makes first-order propagation inadequate.^[ref]