Which operators are number operators of finite families of random variables?

Two sequences of coefficients are introduced for any probability measure on Rd having finite moments of any order. These sequences are natural extensions of the Szegö-Jacobi parameters from the classical one-dimensional case. We review next the definition of the number operator and the fundamental commutation relationships satisfied by the number and quantum operators. The main purpose of this paper is to try to answer the following question: which operators are the number operators of probability measures on Rd having finite moments of any order? There are two types of conditions that these operators must satisfy: algebraic which are expressed in terms of commutators, degrees, and leading coefficients, and analytical which are expressed in terms of positive definiteness of some matrices. Even though we are not able to address the analytical conditions, we find the complete algebraic conditions.