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For stationary continuous detection, let $C_j$ count all cross-channel pairs with $t_2-t_1$ in a bin of full width $\Delta \tau$, and let $r_1,r_2$ be observed single-count rates. Let $T_j$ be their joint live observation time at that delay, approximately the common duration $T_{\mathrm{obs}}$ when delays are short. Independent streams have expected pair count $r_1{r}_2{T}_j\Delta \tau$, giving the displayed estimator. It estimates a bin average of second-order optical coherence, further blurred by detector timing response. For a finite uninterrupted run the exact overlap weighting is $T_{\mathrm{obs}}-|\tau |$ inside the delay integral, rather than a constant at long delays.

For an original numerical example, choose $r_1=2000{\mathrm{s}}^{-1}$, $r_2=3000{\mathrm{s}}^{-1}$, $T_j=100\mathrm{s}$, and $\Delta \tau =10\mathrm{ns}$. The independent reference is six pairs per bin. Chosen counts $C_j=3,6,12$ give estimates $0.5,1,2$; they are teaching inputs with large counting fluctuations, not historical measurements. For pulsed sources, compare properly normalized same-pulse and separate-pulse coincidences instead of assuming this continuous-source baseline.

Detector dead time suppresses nearby same-detector counts and can manufacture an apparent dip in a single-detector record. Two independent detector channels avoid that particular same-channel exclusion, but shared electronics can still reject pairs. Timing spread and wide bins wash out narrow features. A device recording only the first stop after each start measures a waiting-time distribution; interpreting it as all-pair correlations requires a sufficiently low stop probability over the relevant delay range.

Photon antibunching means suppression of close pairs compared with pairs at some later delay: $g^{(2)}(0)<g^{(2)}(\tau )$ for some $\tau$. If correlations decay to the independent-count reference, $g^{(2)}(0)<1$ is a sufficient signature. It violates the classical intensity bounds in photon bunching. For the specified single-mode number measurement, second-order optical coherence also shows that $g^{(2)}(0)<1$ is equivalent to $\mathrm{Var}N<\mu$. This variance criterion concerns that mode or gate; it must not be applied unchanged to arbitrarily long integrated counting intervals.

For one selected polarization of waves, write the detected complex fields as $E_1(t),E_2(t+\tau )$ and their light intensities as $I_j=|E_j{|}^2$, absorbing a common conversion constant. With stationary ensemble averages and nonzero ${\bar{I}}_j=⟨I_j⟩$, defineHere second-order optical coherence concerns distinct detector channels, with the corresponding normal ordering in the quantum formulation. Independent detection has $g^{(2)}=1$. Subtracting the means gives the normalized covarianceThus multiplying raw photocurrents contains a large mean-product contribution; multiplying their fluctuations isolates the small excess. Independent detector shot noise does not supply a mean cross-channel excess, although it increases measurement uncertainty.

The displayed Siegert relation requires a zero-mean, jointly circular complex Gaussian random field, the usual chaotic thermal light model. Circular means invariance under a common arbitrary complex phase rotation; in particular $⟨E_i{E}_j⟩=0$. Apply the Gaussian moment pairing theorem to the real and imaginary parts:The remaining pairing vanishes by circularity. Division proves the Siegert relation. Gaussianity is an assumption about fourth moments, not a consequence of merely measuring first-order optical coherence; a coherent state or photon antibunching cannot be interpreted by this thermal formula.

For equal-intensity beams combined in a common optical mode with matched wave polarizations, interferometric visibility is $V=|g_{12}^{(1)}|$. Consequently ideal thermal stellar intensity interferometry measures $g^{(2)}-1=V^2$, while the full $g^{(2)}$ includes the independent-count reference one. Unpolarized thermal light with two independent equally populated wave polarizations has half this excess: adding their covariances doubles the numerator while multiplying total means quadruples the denominator. Finite detector response further suppresses the peak and must be calibrated.^[ref]