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

## TL;DR

This paper analyzes the convergence rate of the quasi-stationary distribution of the Shiryaev-Roberts diffusion process, establishing an explicit bound that improves understanding of its asymptotic behavior as the boundary grows large.

## Contribution

The paper provides the first explicit bound on the convergence rate of the quasi-stationary distribution for the Shiryaev-Roberts diffusion, using new bounds based on Bessel function properties.

## Key findings

- Convergence rate is no worse than O(log(A)/A) as A→∞.
- Constructed tight bounds for the quasi-stationary cdf using Bessel function properties.
- Results are uniform in x≥0.

## 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).