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Correlated spectral endpoint error means that the errors in two estimated frequencies have nonzero covariance. Their effect on a frequency difference depends on that covariance as well as on the uncertainty of each endpoint. Shared errors can cancel in a difference without making the absolute frequencies accurately known.

Let endpoint errors be ${ϵ}_0,{ϵ}_T$, with standard uncertainties $u_0,u_T$ and correlation coefficient $\rho$. The error in the increment is ${ϵ}_T-{ϵ}_0$, soFor equal uncertainties $u$, the result is $2u^2(1-\rho )$. Independent errors give $2u^2$, perfectly shared errors give zero, and perfect anticorrelation gives $4u^2$. This is correlated-input uncertainty propagation applied to a difference.^[ref]

An original common-offset model writes ${ϵ}_0=b+{\eta }_0$ and ${ϵ}_T=b+{\eta }_T$. Suppose $b$ has variance $s_b^2$ and the two independent residual errors each have variance $s^2$, all mutually independent. Each endpoint then has variance $s_b^2+s^2$ and their covariance is $s_b^2$, butWith $s_b=0.3$ and $s=0.1$ in compatible frequency units, each endpoint has standard uncertainty about $0.316$, while the increment error has standard uncertainty about $0.141$. This spectral common-mode offset rejection follows from subtraction, rather than from knowing the common offset's value.

A shared gain error behaves differently. If $Y_t=(1+g){\omega }_t+b$, the constant offset cancels but $Y_T-Y_0=(1+g)({\omega }_T-{\omega }_0)$. A shared $1\%$ scale error therefore remains a $1\%$ increment error. Calling an error common does not establish which part cancels.

To estimate covariance, repeated measurements should preserve the reference and data-processing relationships relevant to the two endpoints. Filtering or overlapping fitting windows can introduce correlations. The spectral frequency-increment measurement noise contribution also depends on covariance between physical fluctuations and estimation errors; the endpoint formula alone does not justify dropping that separate term. A two-dimensional infrared spectroscopy statistic should be interpreted through its actual estimator and response model, not by automatically adding independent noise to two abstract frequency labels.

The spectral-diffusion frequency-increment variance quantifies how much a transition frequency changes between two observation times. For a real stationary process with mean $\overline{\omega }$ and autocovariance function $C(T)$,The mean increment is zero under these assumptions, so its second moment is also its variance. Expanding the square proves the identity: the two squared-frequency deviations each contribute $C(0)$ and their cross term contributes $-2C(T)$.

If $C(T)={\Delta }^2{e}^{-|T|/\tau }$, thenFor an original example with $\Delta =2\mathrm{rad}\mathrm{p}{\mathrm{s}}^{-1}$ and $T=\tau$, $D=8(1-e^{-1})\simeq 5.057\mathrm{ra}{\mathrm{d}}^2\mathrm{p}{\mathrm{s}}^{-2}$ and the root-mean-square increment is about $2.249\mathrm{rad}\mathrm{p}{\mathrm{s}}^{-1}$. The single-time frequency spread remains $\Delta$; temporal decorrelation does not require the stationary distribution to broaden.

The long-time value $2C(0)$ assumes that the autocovariance function tends to zero. It is not a universal upper bound. For a random-phase sinusoid $\delta \omega (t)=A\cos (\Omega t+\Phi )$ with phase uniform over a full cycle,At half a cycle, the frequencies are anticorrelated and $D=4C(0)$; at a complete cycle the increment vanishes. Thus a nonmonotone increment statistic can reflect recurring frequency memory.

For nonstationary observations $X,Y$, the general identity instead isDrift of the mean must not be silently assigned to stationary spectral diffusion. Likewise, under an explicit model of independent zero-mean endpoint measurement errors with variance $s^2$, the observed second moment gains $2s^2$. This spectral frequency-increment measurement noise term changes when the endpoint errors are correlated.

Primary two-dimensional infrared spectroscopy work uses controlled frequency labeling and later readout to constrain frequency memory.^[ref] The identity above is a model relation; a measured 2D IR center-line slope needs the stated response assumptions before it supplies $C(T)$.