= Power-law tail from a decaying rate envelope
{title2=$P(t)\leq A(1+(t-b)/t_0)^{-\alpha}$}
A \b[power-law tail from a decaying rate envelope] arises when a positive instantaneous decay rate weakens with time. The <time-dependent decay-rate envelope> with $k(t)=\alpha/(t-b+t_0)$ gives
$$
P(t)\leq A\left(1+\frac{t-b}{t_0}\right)^{-\alpha},\qquad t\geq b,
$$
for $A\geq0$, $t_0>0$ and $\alpha>0$. The parent article obtains this envelope by integrating the rate. Its use for recording design additionally requires integrating over the entire unrecorded tail.
For $\alpha>1$, integration after an extension $L\geq0$ gives
$$
U_{\rm after}(L)\leq\frac{At_0}{\alpha-1}
\left(1+\frac L{t_0}\right)^{1-\alpha}.
$$
Hence the <acquisition duration from a power-law tail-energy tolerance> $\varepsilon>0$ is zero when $At_0/(\alpha-1)\leq\varepsilon$; otherwise it is sufficient to choose
$$
L\geq t_0\left[\left(\frac{At_0}{(\alpha-1)\varepsilon}\right)^{1/(\alpha-1)}-1\right].
$$
In an original example, $A=0.10\,\mathrm W$, $t_0=10\,\mathrm{ns}$ and $\alpha=2$ give an initial energy allowance $1.0\,\mathrm{nJ}$. Reducing the remaining allowance to $0.010\,\mathrm{nJ}$ requires $L\geq990\,\mathrm{ns}$. An exponential envelope with the same initial allowance and a $10\,\mathrm{ns}$ decay time would require only $10\log100\approx46.05\,\mathrm{ns}$. Equal initial tail-energy allowances therefore do not imply equal acquisition durations.
For $0<\alpha\leq1$, the envelope tends to zero but its integral after every finite extension diverges. These assumptions then supply no finite missed-energy guarantee. They do not imply that every signal lying below the envelope has infinite energy.
This is an original comparison model, not a measured decay law attributed to a particular detector. In optical pulse metrology, NIST's response measurement separates receiver dynamics from the incident pulse before applying bandwidth corrections.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[Appendix C.1–C.2, printed pp. 36–37]{ref} The same physical distinction is necessary before interpreting any measured slow tail as a global bound on optical power.
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