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.

翻译：我们考虑区间[0,A]上的经典Shiryaev--Roberts鞅扩散过程(R_t)_{t≥0}，其中A>0为预设吸收边界。我们重新审视该扩散在时间极限t→+∞时对每个A>0表现出的拟平稳性（即无吸收条件下的时不变概率行为）这一熟知现象。我们获得了拟平稳分布概率密度函数q_A(x)的新上下界，这些界在简洁性与紧致性之间有所权衡。该界直接推导出预期结果：当A→+∞时，q_A(x)逐点收敛至扩散平稳分布的概率密度函数h(x)（对所有x≥0成立）。该界还给出了固定x处q_A(x)与h(x)之间差距的显式上界。通过积分，q_A(x)的界可转化为对应累积分布函数Q_A(x)的新界。我们利用Polunchenko(2017)近期推导的q_A(x)精确闭式解中修正贝塞尔K函数的最新单调性性质，显式建立了所有结果。最后讨论了本研究成果在最快变点检测中的潜在应用：我们的界可实现对所谓随机化Shiryaev--Roberts--Pollak变点检测过程的高精度性能分析。