  We consider the classical Shiryaev--Roberts martingale diffusion,
$(R_t)_{t\ge0}$, restricted to the interval $[0,A]$, where $A>0$ is a preset
absorbing boundary. We take yet another look at the well-known phenomenon of
quasi-stationarity (time-invariant probabilistic behavior, conditional on no
absorbtion hitherto) exhibited by the diffusion in the temporal limit, as
$t\to+\infty$, for each $A>0$. We obtain new upper- and lower-bounds for the
quasi-stationary distribution's probability density function (pdf), $q_{A}(x)$;
the bounds vary in the trade-off between simplicity and tightness. The bounds
imply directly the expected result that $q_{A}(x)$ converges to the pdf,
$h(x)$, of the diffusion's stationary distribution, as $A\to+\infty$; the
convergence is pointwise, for all $x\ge0$. The bounds also yield an explicit
upperbound for the gap between $q_{A}(x)$ and $h(x)$ for a fixed $x$. By virtue
of integration the bounds for the pdf $q_{A}(x)$ translate into new bounds for
the corresponding cumulative distribution function (cdf), $Q_{A}(x)$. All of
our results are established explicitly, using certain latest monotonicity
properties of the modified Bessel $K$ function involved in the exact
closed-form formula for $q_{A}(x)$ recently obtained by Polunchenko (2017). We
conclude with a discussion of potential applications of our results in quickest
change-point detection: our bounds allow for a very accurate performance
analysis of the so-called randomized Shiryaev--Roberts--Pollak change-point
detection procedure.
