Statistics & Probability Letters

We propose an infinite hidden Markov model for cylindrical time series with von Mises-Gamma emissions. Posterior inference is performed using a beam sampler combining conjugate updates and approximate sampling schemes. Simulation studies and two real data applications demonstrate the effectiveness of the proposed methodology.

Temporal point processes summarize event dependence through factorial moments and cumulants, whereas signature methods preserve event order. We interpret the strict iterated-sums signature through chronological restrictions of factorial moment measures. Abelianisation recovers the probability-generating functional and factorial cumulant projections. At second order, the tensor logarithm separates into a chronological transform of the pair factorial cumulant and an intensity-driven commutator. The latter can remain nonzero for a Poisson process. An explicit two-window Cox example isolates both contributions. The decomposition expresses the second expected-signature cumulant through the first two factorial measures and identifies the chronology baseline of a Poisson process with the same intensity.

A random population of trajectories can be viewed as a point process whose points are entire paths. When the spatial domain contains holes or obstacles, a path also carries global route information, such as how many times it winds around an obstacle. We represent this information by line integrals against a finite family of smooth one-forms supplied by Hodge–Morrey theory; on an annulus, the resulting coordinate is the usual winding number. The main result expresses the cumulant generating function of the aggregate coordinate vector directly in terms of the factorial cumulant measures of the path process. Differentiation yields an all-order partition formula for these topological cumulants. The covariance separates single-trajectory variation from non-Poisson dependence between trajectories. The construction is independent of the chosen closed-form representatives and transforms naturally under a change of coordinate basis. For Poisson path populations the law reduces to compound Poisson, and balanced clockwise and anticlockwise populations give a centred Skellam law.