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

We consider the classical Shiryaev–Roberts martingale diffusion, \((R_t)_{t\geqslant 0}\) , 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 absorption 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 \(x\geqslant 0\) . The bounds also yield an explicit upper bound 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 (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.