The relative-error threshold for extremal Gaussian trace bounds

For the Gaussian trace estimator $T_m$ with $m$ samples and a nonzero real
symmetric positive semidefinite matrix $A$ of effective rank
$\mu = \operatorname{tr}(A)/\|A\|_2$, the manuscript proves that for every
$\varepsilon \ge 2/(m\mu)$ the relative-error tail
$\Pr(|T_m(A) - \operatorname{tr}(A)| \ge \varepsilon \operatorname{tr}(A))$ is
at most the corresponding tail for the extremal matrix
$\mu^{-1}\operatorname{diag}(I_{\lfloor\mu\rfloor}, \mu - \lfloor\mu\rfloor)$,
which in turn is at most $\Pr(|X - 1| \ge \varepsilon)$ for $X$ Gamma
distributed with shape and rate $m\mu/2$; upper and lower tails are compared
separately. The estimator is written as a weighted sum of independent Gamma
variables with weights at most one, and weight is moved between two
fractional coefficients, the smaller always a smallest positive one. A mode
bound for Gamma convolutions, with their unimodality, keeps both tails
monotone along each move, and infinite divisibility gives the final
comparison.