关于Shiryaev-Roberts扩散的拟平稳分布的收敛速度
On the Convergence Rate of the Quasi- to Stationary Distribution for the Shiryaev-Roberts Diffusion
摘要：
对于经典的Shiryaev-Roberts鞅扩散被认为是在区间$[0,A]$，其中$A>0$是一个给定的吸收边界，它表明，扩散的准平稳累积分布函数（CDF），$Q_{A}(x)$，其平稳CDF，$H(x)$，作为$A\to+\infty$的收敛速度不差于$O(\log(A)/A)$，一致在$x\ge0$。通过使用Polunchenko（2017）最近获得的$Q_{A}(x)$的精确闭合形式公式中涉及的修改的Bessel $K$函数的某些最新单调性性质来构造$Q_{A}(x)$的新的紧上下界，明确地建立了结果。
英文摘要：
For the classical Shiryaev--Roberts martingale diffusion considered on the interval $[0,A]$, where $A>0$ is a given absorbing boundary, it is shown that the rate of convergence of the diffusion's quasi-stationary cumulative distribution function (cdf), $Q_{A}(x)$, to its stationary cdf, $H(x)$, as $A\to+\infty$, is no worse than $O(\log(A)/A)$, uniformly in $x\ge0$. The result is established explicitly, by constructing new tight lower- and upper-bounds for $Q_{A}(x)$ 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).
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