The axiomatic classification of natural products in noncommutative probability yields four types of independence: classical (also known as tensor), free, Boolean and monotone; other possibilities, such as Fermionic (i.e., \(\mathbb{Z}_2\)-graded) independence, may occur if an additional structure is available. Monotone independence is distinguished from the others by the noncommutativity of the corresponding natural product; consequently, the correct generalisation of classical notions to the monotone setting is not always clear.
In the work under review, the authors present a theory of generalised cumulants, which agrees with existing theories for classical, free and Boolean independence, but which also includes the monotone case. A family of maps \(\{ K_n : \mathcal{A} \to \mathbb{C} \}_{n \geqslant 1}\) on an algebraic probability space \(( \mathcal{A}, \phi )\) is a family of cumulants for a given type of independence if, for all \(X \in \mathcal{A}\) and \(n \geqslant 1\),
(K1’) \(K_n( X_1 + \cdots + X_N ) = N K_n( X )\) whenever \(X_1, \dots, X_N\) are independent and identically distributed copies of \(X\), i.e., \(\phi( X^k_i ) = \phi( X^k )\) for all \(k \geqslant 1\) and \(i = 1,\dots, N\);
(K2) \(K_n( \lambda X ) = \lambda^n K_n( X )\) for all \(\lambda > 0\);
(K3) \(\phi( X^n ) = K_n( X ) + Q_n\bigl( K_1( X ), \dots, K_{n - 1}( X ) \bigr)\) for some polynomial \(Q_n\) in \(n - 1\) noncommuting variables.
Note that the polynomial \(Q_n\) in (K3) is required to be universal, i.e., independent of \(X\). The axioms (K2) and (K3) are those of F. Lehner [Math. Z. 248, No. 1, 67–100 (2004; Zbl 1089.46040)]; axiom (K1’) is the appropriate substitute for Lehner’s additivity axiom, \[ K_n( X + Y ) = K_n( X ) + K_n( Y ) \quad \text{whenever \(X\) and \(Y\) are independent}, \] which does not make sense in the monotone setting.
The existence and uniqueness of monotone cumulants is established, and then used to give short proofs of monotone-independent versions of the central limit theorem and the law of small numbers.
The authors conclude with the proof of a combinatorial moment-cumulant formula, expressing moments as weighted sums of monotone cumulants over the set of monotone partitions (introduced by Muraki in his classification of quasi-universal products). The analogous formula holds for classical, free and Boolean independence, with monotone partitions replaced by linearly ordered partitions, linearly ordered non-crossing partitions and linearly ordered interval partitions, respectively.