On the quasi-stationary distribution of the Shiryaev–Roberts diffusion. Issue 1 (2nd January 2017) - Polunchenko, Aleksey S.
ABSTRACT: We consider the diffusion generated by the equation with fixed, and where μ ≠0 is given, and is standard Brownian motion. We assume that is stopped at with A >0 preset, and obtain a closed-from formula for the quasi-stationary distribution of, i.e., the limit, x ∈[0, A ]. Further, we also prove Q A ( x ) to be unimodal for any A >0, and obtain its entire moment series. More importantly, the pair with r ≥0 and A >0 is the well-known Generalized Shiryaev–Roberts change-point detection procedure, and its characteristics for r ∼ Q A ( x ) are of particular interest, especially when A >0 is large. In view of this circumstance, we offer an order-three large- A asymptotic approximation of Q A ( x ) valid for all x ∈[0, A ]. The approximation is rather accurate even if A is lower than what would be considered "large" in practice.