On the Convergence Rate of the Quasi- to Stationary Distribution for the Shiryaev-Roberts Diffusion

Kexuan Li; Aleksey S. Polunchenko

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arXiv · 1907.02676

On the Convergence Rate of the Quasi- to Stationary Distribution for the Shiryaev-Roberts Diffusion

Kexuan Li·Aleksey S. Polunchenko

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Abstract

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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Kexuan Li, Aleksey S. Polunchenko. 2019-07-05. On the Convergence Rate of the Quasi- to Stationary Distribution for the Shiryaev-Roberts Diffusion. https://arxiv.org/abs/1907.02676

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