More on the Quasi-Stationary Distribution of the Shiryaev–Roberts Diffusion

We consider the classical Shiryaev–Roberts martingale diffusion, ( R t ) t ⩾ 0 $$(R_t)_{t\geqslant 0}$$ , restricted to the interval [ 0 , A ] $$[0,A]$$ , where A > 0 $$A>0$$ 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 absorption hitherto) exhibited by the diffusion in the temporal limit, as t → + ∞ $$t\to +\infty $$ , for each A > 0 $$A>0$$ . We obtain new upper and lower bounds for the quasi-stationary distribution’s probability density function (pdf), q A ( x ) $$q_{A}(x)$$ ; the bounds vary in the trade-off between simplicity and tightness. The bounds imply directly the expected result that q A ( x ) $$q_{A}(x)$$ converges to the pdf, h ( x ) $$h(x)$$ , of the diffusion’s stationary distribution, as A → + ∞ $$A\to +\infty $$ ; the convergence is pointwise for all x ⩾ 0 $$x\geqslant 0$$ . The bounds also yield an explicit upper bound for the gap between q A ( x ) $$q_{A}(x)$$ and h ( x ) $$h(x)$$ for a fixed x. By virtue of integration, the bounds for the pdf q A ( x ) $$q_{A}(x)$$ translate into new bounds for the corresponding cumulative distribution function (cdf), Q A ( x ) $$Q_{A}(x)$$ . 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 ) $$q_{A}(x)$$ recently obtained by Polunchenko (Sequential Anal 36(1):126–149). 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.