Closed-form distribution of maximum spindle amplitude in the OU model

Derive a closed-form expression or rigorous asymptotic approximation for the probability distribution of the maximum spindle amplitude attained during a segmented spindle generated by the two-dimensional Ornstein–Uhlenbeck process, explicitly in terms of the noise intensity and damping parameters.

Background

The paper develops a series representation for the distribution of spindle maximum amplitude by conditioning on spindle duration and using spectral expansions for first-passage times, but the authors indicate that this approach does not yield a closed-form distribution.

They suggest that a Rayleigh distribution provides a practical approximation for the amplitude statistics observed in simulations, highlighting the need for an exact or asymptotic closed-form result.

References

We proposed a series formula for the distribution of the spindle maximum amplitude, but we could not find a closed form. However, the Rayleigh distribution seemed to be a good approximation.

— Modeling, Segmenting and Statistics of Transient Spindles via Two-Dimensional Ornstein-Uhlenbeck Dynamics (2512.10844 - Sun et al., 11 Dec 2025) in Section 5, Statistical properties of spindles generated by the OU process