  Generalizing the realized variance, the realized skewness (Neuberger, 2012)
and the realized kurtosis (Bae and Lee, 2020), we construct realized cumulants
with the so-called aggregation property. They are unbiased statistics of the
cumulants of a martingale marginal based on sub-period increments of the
martingale and its lower-order conditional cumulant processes. Our key finding
is a relation between the aggregation property and the complete Bell
polynomials. For an application we give an alternative proof and an extension
of a cumulant recursion formula recently obtained by Lacoin et al. (2019) and
Friz et al. (2020).
