L-cumulants, L-cumulant embeddings and algebraic statistics

In this paper we generalize the combinatorial definition of cumulants. Obtained in this way L-cumulants keep all the desired properties of the classical cumulants like semi-invariance and vanishing for independent blocks of random variables. Also the Brillinger's formula for L-cumulants in terms of the conditional L-cumulants is the same as in the classical case. These properties make L-cumulants useful in algebraic analysis of statistical models which we show for general Markov models and hidden Markov processes in the case when the hidden process is binary.