 Validated lower bound on a power-law tail exponent

A validated lower bound on a power-law tail exponent is an exponent constraint that is justified over the future interval where it is used to bound unrecorded energy. A fitted slope over a finite observation window is not by itself such a guarantee. Acquisition duration from a power-law tail-energy tolerance needs a global optical-power envelope, including a lower exponent limit above one for its finite-energy guarantee.

One sufficient condition, for positive absolutely continuous power, isIntegrating the logarithmic derivative gives $P(b+u)\le A(1+u/t_0{)}^{-{\alpha }_{\min }}$. The time-dependent decay-rate envelope provides the equivalent integrating-factor argument when zeros of the power need handling. This is a sufficient condition, not a necessary characterization of every bounded tail.

For an original conditional estimation example, assume an exact power-law family with known $t_0=10\mathrm{ns}$ and initial power $A=0.10\mathrm{W}$. At $u=90\mathrm{ns}$, a measured ratio range $0.008\le P(b+u)/P(b)\le 0.012$ yieldsapproximately $1.9208\le \alpha \le 2.0969$. Within that assumed family, the lower endpoint gives a conservative extension about $123.32\mathrm{ns}$ for a $0.10\mathrm{nJ}$ remaining-energy tolerance. A nominal exponent two would instead require $90\mathrm{ns}$.

The ratio interval constrains the exponent only conditional on the exact family, known time scale and suitable measurement uncertainty. It cannot exclude a later secondary pulse or a transition to a slower law after the observed window. Future-domain validity of a fitted power-law optical envelope therefore requires independent physical justification or explicitly limited claims. If the justified exponent set reaches one, this envelope alone supplies no finite all-time energy guarantee.

Finally, the measured trace must represent optical power rather than an uncorrected receiver tail. NIST measured the detector, cable and oscilloscope together using a short characterized optical input before correcting pulse measurements.^[ref] The exponent example is an original model calculation, not an observed law attributed to that apparatus.