We consider the classical Shiryaev--Roberts martingale diffusion,
(Rt)t≥0, 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→+∞, for each
A>0. We obtain new upper- and lower-bounds for the quasi-stationary distribution's probability density function (pdf),
qA(x); the bounds vary in the trade-off between simplicity and tightness. The bounds imply directly the expected result that
qA(x) converges to the pdf,
h(x), of the diffusion's stationary distribution, as
A→+∞; the convergence is pointwise, for all
x≥0. The bounds also yield an explicit upperbound for the gap between
qA(x) and
h(x) for a fixed
x. By virtue of integration the bounds for the pdf
qA(x) translate into new bounds for the corresponding cumulative distribution function (cdf),
QA(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
qA(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.