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We consider the classical Shiryaev--Roberts martingale diffusion,$(R_t)_{t\ge0}$, restricted to the interval $[0,A]$, where $A>0$ is a presetabsorbing boundary. We take yet another look at the well-known phenomenon ofquasi-stationarity (time-invariant probabilistic behavior, conditional on noabsorbtion 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 thequasi-stationary distribution's probability density function (pdf), $q_{A}(x)$;the bounds vary in the trade-off between simplicity and tightness. The boundsimply directly the expected result that $q_{A}(x)$ converges to the pdf,$h(x)$, of the diffusion's stationary distribution, as $A\to+\infty$; theconvergence is pointwise, for all $x\ge0$. The bounds also yield an explicitupperbound for the gap between $q_{A}(x)$ and $h(x)$ for a fixed $x$. By virtueof integration the bounds for the pdf $q_{A}(x)$ translate into new bounds forthe corresponding cumulative distribution function (cdf), $Q_{A}(x)$. All ofour results are established explicitly, using certain latest monotonicityproperties of the modified Bessel $K$ function involved in the exactclosed-form formula for $q_{A}(x)$ recently obtained by Polunchenko (2017). Weconclude with a discussion of potential applications of our results in quickestchange-point detection: our bounds allow for a very accurate performanceanalysis of the so-called randomized Shiryaev--Roberts--Pollak change-pointdetection procedure.

More On the Quasi-Stationary Distribution of the Shiryaev--Roberts Diffusion